Abstract

Derivations of general equations to describe the absorptance, reflectance, and transmittance of any number of parallel surfaces are presented. Two corollaries describing the propagation of light by two surfaces are derived and then used to derive the general equations. These general equations have wide utility because they describe exactly the propagation of light by any number or type of parallel surfaces (interfaces), using relatively simple expressions, which are readily adaptable for digital computer use.

© 1978 Optical Society of America

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References

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  1. G. G. Stokes, “On the Intensity of the Light Reflected from or Transmitted Through a Pile of Plates,” in Proceedings of the Royal Society of London11, 545 (30 Nov. 1860 to 27 Feb. 1862).
  2. N. W. Kerdub, Gelietekhnika 1 (2), 52 (1(1965).
  3. A. M. Zarem, D. D. Erway, Introduction to Utilization of Solar Energy (McGraw-Hill, New York, 1963).
  4. J. A. Duffie, W. A. Beckman, Solar Energy Thermal Processes (Wiley, New York, 1974).
  5. H. C. Hottel, B. B. Woertz, Trans. Am. Soc. Mech. Eng. 64, 91 (1942).

1965 (1)

N. W. Kerdub, Gelietekhnika 1 (2), 52 (1(1965).

1942 (1)

H. C. Hottel, B. B. Woertz, Trans. Am. Soc. Mech. Eng. 64, 91 (1942).

Beckman, W. A.

J. A. Duffie, W. A. Beckman, Solar Energy Thermal Processes (Wiley, New York, 1974).

Duffie, J. A.

J. A. Duffie, W. A. Beckman, Solar Energy Thermal Processes (Wiley, New York, 1974).

Erway, D. D.

A. M. Zarem, D. D. Erway, Introduction to Utilization of Solar Energy (McGraw-Hill, New York, 1963).

Hottel, H. C.

H. C. Hottel, B. B. Woertz, Trans. Am. Soc. Mech. Eng. 64, 91 (1942).

Kerdub, N. W.

N. W. Kerdub, Gelietekhnika 1 (2), 52 (1(1965).

Stokes, G. G.

G. G. Stokes, “On the Intensity of the Light Reflected from or Transmitted Through a Pile of Plates,” in Proceedings of the Royal Society of London11, 545 (30 Nov. 1860 to 27 Feb. 1862).

Woertz, B. B.

H. C. Hottel, B. B. Woertz, Trans. Am. Soc. Mech. Eng. 64, 91 (1942).

Zarem, A. M.

A. M. Zarem, D. D. Erway, Introduction to Utilization of Solar Energy (McGraw-Hill, New York, 1963).

Gelietekhnika (1)

N. W. Kerdub, Gelietekhnika 1 (2), 52 (1(1965).

Trans. Am. Soc. Mech. Eng. (1)

H. C. Hottel, B. B. Woertz, Trans. Am. Soc. Mech. Eng. 64, 91 (1942).

Other (3)

G. G. Stokes, “On the Intensity of the Light Reflected from or Transmitted Through a Pile of Plates,” in Proceedings of the Royal Society of London11, 545 (30 Nov. 1860 to 27 Feb. 1862).

A. M. Zarem, D. D. Erway, Introduction to Utilization of Solar Energy (McGraw-Hill, New York, 1963).

J. A. Duffie, W. A. Beckman, Solar Energy Thermal Processes (Wiley, New York, 1974).

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Figures (3)

Fig. 1
Fig. 1

Light from an internal source propagated by two surfaces.

Fig. 2
Fig. 2

Light from an external source propagated by two surfaces.

Fig. 3
Fig. 3

Light from an external source propagated by N surfaces.

Equations (36)

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a n = exp [ KL sec ( i ) ] ,
b n sin ( i ) = b n 1 sin ( i ) ,
r n = 1 2 [ sin 2 ( i i ) sin 2 ( i + i ) + tan 2 ( i i ) tan 2 ( i + i ) ] ,
I r i = I r n + 1 a n 2 ( 1 r n ) + I r n r n + 1 2 a n 4 ( 1 r n ) + I r n 2 r n + 1 3 a n 6 ( 1 r n ) + I r n 3 r n + 1 4 a n 8 ( 1 r n ) + . . . ,
I r i = I r n + 1 a n 2 ( 1 r n ) ( 1 + r n r n + 1 a n 2 + r n 2 r n + 1 2 a n 4 + r n 3 r n + 1 3 a n 6 + . . . ) .
I r i = I r n + 1 a n 2 ( 1 r n ) ( 1 r n r n + 1 a n 2 )
r i = I r n + 1 a n 2 ( 1 r n ) ( 1 r n r n + 1 a n 2 ) .
I t i = I a n ( 1 r n + 1 ) + ( I r n r n + 1 a n 3 ( 1 r n + 1 ) + I r n 2 r n + 1 2 a n 5 ( 1 r n + 1 ) + I r n 3 r n + 1 3 a n 7 ( 1 r n + 1 ) + . . . ,
I t i = I a n ( 1 r n + 1 ) + ( 1 + r n r n + 1 a n 2 + r n 2 r n + 1 2 s n 4 + r n 3 r n + 1 3 a n 6 + . . . ) .
I t i = I a n ( 1 r n + 1 ) ( 1 r n r n + 1 a n 2 ) .
t i = a n ( i r n + 1 ) ( 1 r n r n + 1 a n 2 ) .
r i = r n + 1 a n 2 ( 1 r n ) ( 1 r n r n + 1 a n 2 )
t i = a n ( 1 r n + 1 ) ( 1 r n r n + 1 a n 2 ) .
I r e = I r n + I r n + 1 a n 2 ( 1 r n ) I r n + 1 r n a n 2 ( 1 r n ) + I r n + 1 2 r n a n 4 ( 1 r n ) I r n + 1 2 r n 2 a n 4 ( 1 r n ) + . . . ,
I r e = I r n + I ( 1 r n ) 2 r n ( r n + 1 r n a n 2 + r n + 1 2 r n 2 a n 4 + . . . ) .
I r e = I r n + I ( 1 r n 2 ) r n ( 1 1 r n r n + 1 a n 2 1 ) .
I r e = I r n + I ( 1 r n 2 ) 2 a n 2 r n + 1 ( 1 a n 2 r n r n + 1 )
r e = r n + ( 1 r n ) 2 a n 2 r n + 1 ( 1 a n 2 r n r n + 1 ) .
r e = r n + ( 1 r n ) 2 a n 2 r n 1 a n 2 r n 2
I t e = I [ a n ( 1 r n ) r n + 1 a n ( 1 r n ) ] + I [ r n + 1 r n a n 3 ( 1 r n ) r n + 1 3 r n 2 a n 3 ( 1 r n ) ] + I [ r n + 1 2 r n 2 a n 5 ( 1 r n ) r n + 1 3 r n 2 a n 5 ( 1 r n ) ] + I [ r n + 1 3 r n 3 a n 7 ( 1 r n ) r n + 1 4 r n 3 a n 7 ( 1 r n ) ] + . . . .
I t e = I a n ( 1 r n ) ( 1 r n + 1 ) [ 1 a n 2 r n + 1 r n + a n 4 r n + 1 2 r n + a n 6 r n + 1 3 r n 3 + . . . .
I t e = I a n ( 1 r n ) ( 1 r n + 1 ) 1 a n 2 r n + 1 r n .
t e = a n ( 1 r n ) ( 1 r n + 1 ) 1 a n 2 r n + 1 r n .
t e = a n ( 1 r n ) 2 1 a n 2 r n 2 ,
r e = r n + ( 1 r n ) 2 a n 2 r n + 1 ( 1 a n 2 r n r n + 1 ) ,
t e = a n ( 1 r n ) ( 1 r n + 1 ) 1 a n 2 r n + 1 r n .
I T n 1 r i = I T n 1 r n a r 1 2 ( 1 r n 1 ) ( 1 a r 1 2 r n r n 1 )
I T n 1 t i = I T n 1 a n 1 ( 1 r n ) ( 1 a r 1 2 r n r n 1 )
I T n 1 a n 1 2 ( 1 r n 1 ) a n 2 ( 1 a n 1 2 r n r n 1 )
I T B ( n 2 ) T n 1 r n a n 1 2 a n 2 ( 1 r n 1 ) ( 1 a n 1 2 r n r n 1 )
I R B ( n 2 ) T n 1 r n a n 2 ( 1 r n 1 ) a n 1 2 ( 1 a n 1 2 r n r n 1 )
I R n = I R n 1 + I T B ( n 2 ) T n 1 r n a n 1 2 a n 2 ( 1 r n 1 ) ( 1 a n 1 2 r n r n 1 ) + I T n 1 T B ( n 2 ) R B ( n 2 ) r n a n 1 2 a n 2 3 ( 1 r n 1 ) ( 1 a n 1 2 r n r n 1 ) × [ r n 1 + ( 1 r n 1 ) 2 r n a n 1 2 ( 1 a n 1 2 r n r n 1 ) ] + I T n 1 T B ( n 2 ) R B 2 ( n 2 ) r n a n 1 2 a n 2 5 ( 1 r n 1 ) ( 1 a n 1 2 r n r n 1 ) × [ r n 1 + ( 1 r n 1 ) 2 r n a n 1 2 ( 1 a n 1 2 r n r n 1 ) ] 2 + I T n 1 T B ( n 2 ) R B 3 ( n 2 ) r n a n 1 2 a n 2 7 ( 1 r n 1 ) ( 1 a n 1 2 r n r n 1 ) × [ r n 1 + ( 1 r n 1 ) 2 r n a n 1 2 ( 1 a n 1 2 r n r n 1 ) ] 2 + . . . ,
R n = R n 1 + T n 1 T B ( n 2 ) r n a n 2 a n 1 2 ( 1 r n ) ( 1 a n 1 2 r n r n 1 ) { 1 R B ( n 2 ) a n 2 2 [ r n 1 + ( 1 r n 1 ) 2 r n a n 1 2 ( i a n 1 2 r n r n 1 ) ] } .
I T n = I T n 1 a n 1 ( 1 r n ) ( 1 a n 1 r n r n 1 ) + I T n 1 R B ( n 2 ) r n a n 1 2 a n 2 2 ( 1 r n 1 ) 2 ( i r n ) a n 1 ( 1 a n 1 2 r n r n 1 ) ( 1 a n 1 2 r n r n 1 ) + I T n 1 R 2 B ( n 2 ) r n a n 1 2 a n 2 2 ( 1 r n 1 ) ( 1 a n 1 2 r n r n 1 ) × [ r n 1 + ( 1 r n 1 ) 2 a n 1 2 r n ( 1 a n 1 2 r n r n 1 ) ] ( 1 r n ) ( 1 r n 1 ) a n 1 ( 1 a n 1 2 r n r n 1 ) + . . . ,
T n = T n 1 a n 1 ( 1 r n ) ( 1 a n 1 2 r n r n 1 ) ( 1 + R B ( n 2 ) r n a n 1 2 a n 2 2 ( 1 r n 1 ) 2 ( 1 a n 1 2 r n r n 1 ) { 1 R B ( n 2 ) a n 2 2 [ r n 1 + ( 1 r n 1 ) 2 a n 1 2 r n ( 1 a n 1 2 r n r n 1 ) ] } ) .
A n = T n + 1 [ ( 1 a n 2 r n r n + 1 ) a n ( 1 r n + 1 ) r n + 1 a n ( 1 r n ) ( 1 r n + 1 ) 1 ] ,

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