Abstract

A differential equation is obtained for a reflecting surface that will distribute radiation originating from a linear array of discrete sources or a continuous line source into a specified illuminance over an arbitrary receiver surface that is symmetric about the line. The equation is solved numerically for the special case of two discrete emitting elements and also for a continuous Lambertian source.

© 1975 Optical Society of America

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References

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  1. J. S. Schruben, J. Opt. Soc. Am. 64, 55 (1974).
    [CrossRef]
  2. D. G. Burkhard, D. L. Shealy, Abstract. Opt. Soc. Am.6310, 1296 (Rochester, N.Y., 1973).
  3. T. E. Horton, J. H. McDermit, J. Heat Trans. Trans ASME C94 (4), 453 (1972);Appl. Opt. 13, 1445 (1974).
    [CrossRef]
  4. J. H. McDermit, T. E. Horton, Abstract, J. Opt. Soc. Am. 64, 1357 (1974).
  5. J. H. McDermit, T. E. Horton, Abstract, J. Opt. Soc. Am. 64, 1357 (1974).
  6. D. G. Burkhard, D. L. Shealy, Abstract, J. Opt. Soc. Am. 64, 1357 (1974).
  7. D. G. Burkhard, D. L. Shealy, R. U. Sexl, Int. J. Heat Mass Transfer 16, 271 (1973).
    [CrossRef]
  8. D. L. Shealy, D. G. Burkhard, Int. J. Heat Mass Transfer 15, 281 (1973).
    [CrossRef]
  9. D. G. Burkhard, D. L. Shealy, J. Opt. Soc. Am. 63, 299 (1973).
    [CrossRef]
  10. D. L. Shealy, D. G. Burkhard, Opt. Acta 20, 287 (1973).
    [CrossRef]
  11. D. L. Shealy, D. G. Burkhard, Appl. Opt. 12, 2955 (1973).
    [CrossRef] [PubMed]

1974

J. H. McDermit, T. E. Horton, Abstract, J. Opt. Soc. Am. 64, 1357 (1974).

J. H. McDermit, T. E. Horton, Abstract, J. Opt. Soc. Am. 64, 1357 (1974).

D. G. Burkhard, D. L. Shealy, Abstract, J. Opt. Soc. Am. 64, 1357 (1974).

J. S. Schruben, J. Opt. Soc. Am. 64, 55 (1974).
[CrossRef]

1973

D. G. Burkhard, D. L. Shealy, R. U. Sexl, Int. J. Heat Mass Transfer 16, 271 (1973).
[CrossRef]

D. L. Shealy, D. G. Burkhard, Int. J. Heat Mass Transfer 15, 281 (1973).
[CrossRef]

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

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

D. L. Shealy, D. G. Burkhard, Appl. Opt. 12, 2955 (1973).
[CrossRef] [PubMed]

1972

T. E. Horton, J. H. McDermit, J. Heat Trans. Trans ASME C94 (4), 453 (1972);Appl. Opt. 13, 1445 (1974).
[CrossRef]

Burkhard, D. G.

D. G. Burkhard, D. L. Shealy, Abstract, J. Opt. Soc. Am. 64, 1357 (1974).

D. G. Burkhard, D. L. Shealy, R. U. Sexl, Int. J. Heat Mass Transfer 16, 271 (1973).
[CrossRef]

D. L. Shealy, D. G. Burkhard, Int. J. Heat Mass Transfer 15, 281 (1973).
[CrossRef]

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

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

D. L. Shealy, D. G. Burkhard, Appl. Opt. 12, 2955 (1973).
[CrossRef] [PubMed]

D. G. Burkhard, D. L. Shealy, Abstract. Opt. Soc. Am.6310, 1296 (Rochester, N.Y., 1973).

Horton, T. E.

J. H. McDermit, T. E. Horton, Abstract, J. Opt. Soc. Am. 64, 1357 (1974).

J. H. McDermit, T. E. Horton, Abstract, J. Opt. Soc. Am. 64, 1357 (1974).

T. E. Horton, J. H. McDermit, J. Heat Trans. Trans ASME C94 (4), 453 (1972);Appl. Opt. 13, 1445 (1974).
[CrossRef]

McDermit, J. H.

J. H. McDermit, T. E. Horton, Abstract, J. Opt. Soc. Am. 64, 1357 (1974).

J. H. McDermit, T. E. Horton, Abstract, J. Opt. Soc. Am. 64, 1357 (1974).

T. E. Horton, J. H. McDermit, J. Heat Trans. Trans ASME C94 (4), 453 (1972);Appl. Opt. 13, 1445 (1974).
[CrossRef]

Schruben, J. S.

Sexl, R. U.

D. G. Burkhard, D. L. Shealy, R. U. Sexl, Int. J. Heat Mass Transfer 16, 271 (1973).
[CrossRef]

Shealy, D. L.

D. G. Burkhard, D. L. Shealy, Abstract, J. Opt. Soc. Am. 64, 1357 (1974).

D. G. Burkhard, D. L. Shealy, R. U. Sexl, Int. J. Heat Mass Transfer 16, 271 (1973).
[CrossRef]

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

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

D. L. Shealy, D. G. Burkhard, Int. J. Heat Mass Transfer 15, 281 (1973).
[CrossRef]

D. L. Shealy, D. G. Burkhard, Appl. Opt. 12, 2955 (1973).
[CrossRef] [PubMed]

D. G. Burkhard, D. L. Shealy, Abstract. Opt. Soc. Am.6310, 1296 (Rochester, N.Y., 1973).

Abstract, J. Opt. Soc. Am.

J. H. McDermit, T. E. Horton, Abstract, J. Opt. Soc. Am. 64, 1357 (1974).

J. H. McDermit, T. E. Horton, Abstract, J. Opt. Soc. Am. 64, 1357 (1974).

D. G. Burkhard, D. L. Shealy, Abstract, J. Opt. Soc. Am. 64, 1357 (1974).

Appl. Opt.

Int. J. Heat Mass Transfer

D. G. Burkhard, D. L. Shealy, R. U. Sexl, Int. J. Heat Mass Transfer 16, 271 (1973).
[CrossRef]

D. L. Shealy, D. G. Burkhard, Int. J. Heat Mass Transfer 15, 281 (1973).
[CrossRef]

J. Heat Trans. Trans ASME

T. E. Horton, J. H. McDermit, J. Heat Trans. Trans ASME C94 (4), 453 (1972);Appl. Opt. 13, 1445 (1974).
[CrossRef]

J. Opt. Soc. Am.

Opt. Acta

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

Other

D. G. Burkhard, D. L. Shealy, Abstract. Opt. Soc. Am.6310, 1296 (Rochester, N.Y., 1973).

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Figures (8)

Fig. 1
Fig. 1

Two emitting elements I0 and I1 with reflector surface S1 and receiver surface S2.

Fig. 2
Fig. 2

Reflector that uniformly illuminates a flat collector surface that is perpendicular to the z axis; the base of the collector of radius R is located at z = −2R. The surfaces are axially symmetric about half of the z axis.

Fig. 3
Fig. 3

Continuous line source-with reflector and receiver surface.

Fig. 4
Fig. 4

Scale drawing of uniformly backlighted surface with line source of length R/2. Initial and terminal x,z coordinates of reflector are, respectively, 1.73R, 6.89R and 3.40R, 5.88R.

Fig. 5
Fig. 5

Figure explaining notation of Eq. (18) for calculating total flux emitted by line source of length d; part of the flux is intercepted by the reflector surface, and part is emitted directly to the collector.

Fig. 6
Fig. 6

Scale drawing of uniformly illuminated receiver surface receiving direct illumination and reflected illumination. Receiverradius R, at z = −2R. Line source of length −R/4. Initial and terminal reflector coordinates are R, 2R and 1.06R, 1.84R, respectively.

Fig. 7
Fig. 7

Figure showing cone angles to be used when calculating total flux Et required in Eq. (20).

Fig. 8
Fig. 8

Section of cylindrical reflector that uniformly illuminates the receiver base of radius R with center at z = 4R. Length of source is R/2. Terminal coordinates x,z of reflector are 1.27R, 1.27R.

Equations (51)

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E ( X ) d S 2 = σ μ cos φ i d S 1 .
d S 1 = 2 π R ( R + R 2 ) 1 / 2 sin θ d θ ,
R = dR / d θ .
d S 2 = 2 π X [ 1 + ( dX / dz ) 2 ] 1 / 2 dX .
d S 2 = 2 π XdX .
a = I sin θ + K cos θ .
N = ( z I + K ) / ( 1 + z 2 ) 1 / 2 ,
z = dz / dx .
z = ( R cos θ R sin θ ) / ( R sin θ + R cos θ ) .
cos ϕ i ( N · a ) = R / ( R 2 + R 2 ) 1 / 2 .
σ = ( I 0 sin θ ) / R 2 ,
E ( X ) XdX = I 0 μ sin 2 θ d θ .
0 X E ( X ) XdX = I 0 μ ( θ 2 sin θ cos θ 2 ) I 0 μ ( θ 0 2 sin θ 0 cos θ 0 2 ) ,
A = 2 ( a · N ) N a .
X x Z 0 z = A x A z ,
X = R sin θ + ( d 0 + R cos θ ) 2 R R cos θ ( R 2 R 2 ) 1 / 2 sin θ 2 R R sin θ + ( R 2 R 2 ) 1 / 2 cos θ .
R R = ( X sin θ d 0 cos θ R ) + [ ( X sin θ d 0 cos θ R ) 2 + ( X cos θ + d 0 sin θ ) 2 ] 1 / 2 ( X cos θ + d 0 sin θ ) .
E ( X ) = I 0 μ sin 2 θ X ( dX / d θ ) .
E tot ( X ) = E dir ( X ) + E refl ( X ) .
E refl = E t E dir ,
2 π E d XdX = 2 π I 0 sin 2 θ d θ ,
E d = I 0 sin θ cos 3 θ = I 0 X d 0 / ( d 0 2 + X 2 ) 2 .
E ( X ) = E t I 0 X d 0 / d 0 2 + X 2 ) 2 .
0 X ξ ( X ) XdX + 0 X 1 ξ ( X 1 ) X 1 d X 1 = I 0 ( θ 2 sin θ cos θ 2 ) + I 1 ( θ 1 2 sin θ 1 cos θ 1 2 ) .
tan θ 1 = R sin θ d + R cos θ ,
E ( X ) = E t I 0 X d 0 / ( d 0 2 + X 2 ) 2 I 1 X d 1 / ( d 1 2 + X 2 ) 2 .
0 X E ( X ) XdX = [ ( θ sin θ cos θ ) + ( θ 1 sin θ 1 cos θ 1 ) ] / 2 .
E = E t E d ,
E d = I 0 d 0 X ( d 0 2 + X 2 ) 2 + I 1 ( d 0 d 1 ) X [ ( d 0 d 1 ) 2 + X 2 ] 2 ,
0 X E ( X ) XdX = E t X 2 + I 0 X d 0 / ( d 0 2 + X 2 ) + I 1 X ( d 0 d 1 ) / [ ( d 0 d 1 ) 2 + X 2 ] ( tan 1 X / d 0 ) / d 0 [ tan 1 X / ( d 0 d 1 ) ] / ( d 0 d 1 ) .
sin ( θ θ z ) sin θ z = z R ( θ ) .
d z = R ( θ ) sin θ sin 2 θ z d θ z .
0 X E ( X ) XdX = μ B 0 2 θ 1 θ ( θ z sin θ z cos θ z ) R ( θ ) sin θ sin 2 θ z d θ z μ B 0 2 θ 0 1 θ 0 ( θ z sin θ z cos θ z ) R ( θ 0 ) sin θ 0 sin 2 θ z d θ z .
X 2 = μ B 0 R ( θ ) sin θ ( θ 1 cot θ 1 θ cot θ ) / E t μ B 0 R ( θ 0 sin θ 0 ( θ 01 cot θ 01 θ 0 cot θ 0 ) / E t .
E t = μ B 0 R ( θ m ) sin θ m ( θ 1 m cot θ 1 m θ m cot θ m ) / X m 2 μ B 0 R ( θ 0 sin θ 0 ( θ 01 cot θ 01 θ 0 cot θ 0 ) / X m 2 .
E d = B 0 0 d ( d 0 z ) Xdz [ ( d 0 z ) 2 + X 2 ] 2 ,
E d = B 0 X { 1 2 [ ( d 0 d ) 2 + X 2 ] 1 2 ( d 0 2 + X 2 ) } .
0 X E ( X ) XdX = 0 X ( E t E d ) XdX = 0 X E t XdX 0 X E d XdX .
0 X E t XdX = E t X 2 / 2 .
0 X E d XdX = B 0 2 0 X X 2 dX X 2 + ( d 0 d ) 2 0 X X 2 dX ( X 2 + d 0 2 ) ,
0 X E d XdX = B 0 2 [ d 0 tan 1 X d 0 ( d 0 d ) tan 1 X ( d 0 d ) ] .
E X 2 = E t X 2 B 0 [ d 0 tan 1 ( X d 0 ) ( d 0 d ) tan 1 ( X ( d 0 d ) ) ] .
E t = B 0 [ μ R ( θ m ) sin θ m ( θ 1 m cot θ 1 m θ m cot θ m ) + ρ ( ϕ m ) sin ϕ m ( ϕ 1 m cot ϕ 1 m ϕ m cot ϕ m ) R ( θ 1 ) sin θ 0 ( θ 01 cot θ 01 θ 0 cot θ 0 ) ] / X m 2 .
E d = B 0 0 d ( d 0 + z ) Xdz [ ( d 0 + z ) 2 + X 2 ] 2 ,
E X 2 = E t X 2 B 0 [ ( d 0 + d ) tan 1 X ( d 0 + d ) d 0 tan 1 X d 0 ] ,
0 X max E ( X ) XdX = θ 0 θ max I 0 μ sin 2 θ d θ .
0 X E ( X ) XdX = θ 0 θ I 0 μ sin 2 θ d θ ,
R = R ( θ ) .
0 X E ( X ) XdX = 0 θ I 0 μ sin 2 θ d θ + 0 θ I 0 μ sin 2 θ 1 d θ 1 .
0 X max E ( X ) XdX = 0 θ max I 0 μ sin 2 θ d θ + 0 θ 1 max I 0 μ sin 2 θ 1 d θ 1 .
R = R ( θ ) .

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