Abstract

Expressions for the reflectance of finite, perfect diffuse, and perfect specular samples measured in an integrating sphere wlth finite ports and a uniform, perfectly diffuse reflecting wall, are derived by solving a set of linear equations. The derivation is easy compared with the long computations necessary when using an integral equation approach.

Expressions are given both for the comparison and the substitution method of measurement, with curved or flat sample and standard. For finite, flat, diffuse, and specular samples, computed examples indicate the range of sample sizes within which previous approximate expressions may be used, without introducing significant errors.

© 1965 Optical Society of America

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References

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  1. R. Ulbricht, Das Kugel photometer (R. Oldenburg, Munich and Berlin, 1920).
  2. A. H. Tavlor, Sci. Papers Natl. Bur. Std. (U. S.), No. 391, 421 (1920).
  3. J. S. Preston, Trans. Opt. Soc. (London) 31, 15 (1929–30).
    [CrossRef]
  4. A. C. Hardy and O. W. Pineo, J. Opt. Soc. Am. 21, 502 (1931).
    [CrossRef]
  5. P. Moon, J. Opt. Soc. Am. 30, 195 (1940).
    [CrossRef]
  6. J. A. Jacquez and H. F. Kuppenheim, J. Opt. Soc. Am. 45, 460 (1955).
    [CrossRef]
  7. P. F. O’Brien, J. Opt. Soc. Am. 46, 343 (1956).
    [CrossRef]
  8. P. F. O’Brien and J. A. Howard, Illum. Engr. 54, 177 (1959).
  9. P. F. O’Brien and J. A. Howard, Illum. Engr. 54, 209 (1959).
  10. M. Jakob, Heat Transfer, Vol. II (John Wiley & Sons Inc., New York, 1957).
  11. P. Moon, Scientific Basis of Illumination Engineering (McGraw-Hill Book Co., Inc., New York, 1936).

1959 (2)

P. F. O’Brien and J. A. Howard, Illum. Engr. 54, 177 (1959).

P. F. O’Brien and J. A. Howard, Illum. Engr. 54, 209 (1959).

1956 (1)

1955 (1)

1940 (1)

1931 (1)

1920 (1)

A. H. Tavlor, Sci. Papers Natl. Bur. Std. (U. S.), No. 391, 421 (1920).

Hardy, A. C.

Howard, J. A.

P. F. O’Brien and J. A. Howard, Illum. Engr. 54, 177 (1959).

P. F. O’Brien and J. A. Howard, Illum. Engr. 54, 209 (1959).

Jacquez, J. A.

Jakob, M.

M. Jakob, Heat Transfer, Vol. II (John Wiley & Sons Inc., New York, 1957).

Kuppenheim, H. F.

Moon, P.

P. Moon, J. Opt. Soc. Am. 30, 195 (1940).
[CrossRef]

P. Moon, Scientific Basis of Illumination Engineering (McGraw-Hill Book Co., Inc., New York, 1936).

O’Brien, P. F.

P. F. O’Brien and J. A. Howard, Illum. Engr. 54, 209 (1959).

P. F. O’Brien and J. A. Howard, Illum. Engr. 54, 177 (1959).

P. F. O’Brien, J. Opt. Soc. Am. 46, 343 (1956).
[CrossRef]

Pineo, O. W.

Preston, J. S.

J. S. Preston, Trans. Opt. Soc. (London) 31, 15 (1929–30).
[CrossRef]

Tavlor, A. H.

A. H. Tavlor, Sci. Papers Natl. Bur. Std. (U. S.), No. 391, 421 (1920).

Ulbricht, R.

R. Ulbricht, Das Kugel photometer (R. Oldenburg, Munich and Berlin, 1920).

Illum. Engr. (2)

P. F. O’Brien and J. A. Howard, Illum. Engr. 54, 177 (1959).

P. F. O’Brien and J. A. Howard, Illum. Engr. 54, 209 (1959).

J. Opt. Soc. Am. (4)

Sci. Papers Natl. Bur. Std. (U. S.) (1)

A. H. Tavlor, Sci. Papers Natl. Bur. Std. (U. S.), No. 391, 421 (1920).

Trans. Opt. Soc. (London) (1)

J. S. Preston, Trans. Opt. Soc. (London) 31, 15 (1929–30).
[CrossRef]

Other (3)

R. Ulbricht, Das Kugel photometer (R. Oldenburg, Munich and Berlin, 1920).

M. Jakob, Heat Transfer, Vol. II (John Wiley & Sons Inc., New York, 1957).

P. Moon, Scientific Basis of Illumination Engineering (McGraw-Hill Book Co., Inc., New York, 1936).

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Figures (2)

F. 1
F. 1

Integrating-sphere measuring methods: (a) Comparison method; (b) substitution method. (1) Integrating sphere wall; (2) sample port; (3) standard port; (4) photocell port; (5) entrance port.

F. 2
F. 2

Integrating sphere with specular sample. The ports are numbered as in Fig. 1. The number 6 indicates the position of the area of the sphere wall illuminated by the incident beam after reflection from the specular sample. The primed numbers refer to the sphere imaged in the specular sample (dashed circle).

Tables (3)

Tables Icon

Table I Sample reflectance r2 in the perfect diffuse case, r1 = reflectance of sphere wall, r2 = reflectance of sample (s). r3 = reflectance of standard (st), L1 = luminous emittance of sphere wall, and F = form factor [see Eqs. (3.3), (4.1)].

Tables Icon

Table II The reflectance r2 of diffuse samples computed from the expressions of Table I, with L1s/L1st=0.5, r1 = r3 = 0.97, A4 = A5 = 0.01A, and values of A1, A2, and A2c as shown.

Tables Icon

Table III Reflectance of perfect specular samples, computed from expressions (5.7), (5.6a), and (5.6b), with L1s/L1st = 0.5, r1= r3 = 0.97, A4 = A5 = 0.01A, A2c = 0.01 − 0.03 − 0.05 − 0.10, and values of A1 as shown.

Equations (40)

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A 1 F 12 = A 2 F 21 .
L 1 = r 1 L 1 F 11 + r 1 L 2 F 12 + r 1 E 1 .
L 1 = r 1 L 1 F 11 + r 1 L 2 F 12 + + r 1 L n F 1 n + r 1 E 1 .
[ r 1 F 11 1 r 1 F 12 r 1 F 13 r 1 F 1 n r 2 F 21 r 2 F 22 1 r 2 F 23 r 2 F 2 n r 3 F 31 r 3 F 32 r 3 F 33 1 r 3 F 3 n r n F n 1 r n F n 2 r n F n 3 r n F n n 1 ] [ L 1 L 2 L 3 L n ] = [ r 1 E 1 r 2 E 2 r 3 E 3 r n E n ] .
A 1 = A A 2 A 3 A 4 A 5
A 1 = A A 2 A 4 A 5 .
[ r 1 F 11 1 r 1 F 12 r 1 F 13 r 2 F 21 r 2 F 22 1 r 2 F 23 r 3 F 31 r 3 F 32 r 3 F 33 1 ] [ L 1 L 2 L 3 ] = Sample illuminated Standard illuminated [ 0 r 2 E 0 0 ] [ 0 0 r 3 E 0 ] .
[ r 1 F 11 1 r 1 F 12 r 2 F 21 r 2 F 22 1 ] [ L 1 L 2 ] = [ 0 r 0 E 2 ] .
F 11 = A 1 / A , F 12 = F 13 = A 2 / A , F 21 = F 31 = A 1 / A , F 23 = F 32 = A 2 / A , F 22 = F 33 = A 2 / A .
F 11 = A 1 / A , F 12 = F 13 = A 2 c / A , F 21 = F 31 = A 1 / ( A A 2 c ) , F 23 = F 32 = A 2 c / ( A A 2 c ) .
E 1 = r 1 r 2 E 0 ( A 2 fl / A ) [ 1 + r 1 r 2 ( F 62 F 6 5 ) ] .
E 3 = E 1 ( A 2 c / A 2 fl ) .
[ r 1 A 1 A 1 r 1 A 2 fl A A 2 c r 1 A 1 A A 2 c r 2 A 1 A A 2 c A 2 fl 1 r 2 A 2 fl A A 2 c A 2 c A 2 fl r 3 A 1 A A 2 c A 2 fl r 3 A 2 fl A A 2 c A 2 c A 2 fl 1 ] [ L 1 L 2 L 3 ] = Sample illuminated Standard illuminated [ r 1 E 1 0 r 3 E 1 A 2 c A 2 fl ] [ 0 0 r 3 E 0 ] .
L 1 s L 1 st = [ r 1 r 2 ( A 2 fl / A 2 c ) ] [ 1 + r 1 r 2 ( F 62 F 6 5 ) ] { 1 + r 3 [ A 2 c / ( A A 2 c ) ] } r 3 { 1 + r 2 [ A 2 c / ( A A 2 c ) ] } ,
r 2 = r 3 ( L 1 s / L 1 st ) r 1 ( A 2 fl / A 2 c ) ] [ 1 + r 1 r 2 ( F 62 F 6 5 ) ] + r 3 [ A 2 c / ( A A 2 c ) ] { r 1 ( A 2 fl / A 2 c ) [ 1 + r 1 r 2 ( F 62 F 6 5 ) ] L 1 s / L 1 st } .
[ r 1 A 1 A 1 r 1 A 2 fl A A 2 c r 2 A 1 A A 2 c A 2 fl 1 ] [ L 1 L 2 ] = Sample illum . Stand. illum [ r 1 E 1 0 ] [ 0 r 3 E 0 ] ,
L 1 s L 1 st = r 1 r 2 r 3 ( A 2 fl / A 2 c ) [ 1 + r 1 r 2 ( F 62 F 6 5 ) ] { 1 r 1 ( A 1 / A ) ( 1 + r 3 [ A 2 c / ( A A 2 c ) ] ) } { 1 r 1 ( A 1 / A ) ( 1 + r 2 [ A 2 c / ( A A 2 c ) ] ) } ,
r 2 = r 3 ( L 1 s / L 1 st ) [ 1 r 1 ( A 1 / A ) ] r 1 A 2 fl A 2 c [ 1 + r 1 r 2 ( F 62 F 6 5 ) ] ( 1 r 1 A 1 A ) r 1 r 3 A 1 A A 2 c A A 2 c ( r 1 A 2 fl A 2 c [ 1 + r 1 r 2 ( F 62 F 6 5 ) ] L 1 st L 1 st ) .
r 2 = r 3 ( L 1 s / L 1 st ) r 1 + r 3 [ A 2 c / ( A A 2 c ) ] [ r 1 ( L 1 s / L 1 st ) ] ,
r 2 = r 3 ( L 1 s / L 1 st ) [ 1 r 1 ( A 1 / A ) ] r 1 [ 1 r 1 ( A 1 / A ) ] r 1 r 3 ( A 1 / A ) [ A 2 c / ( A A 2 c ) ] [ r 1 ( L 1 s / L 1 st ) ] .
r 2 = ( r 3 / r 1 ) ( L 1 s / L 1 st ) .
L 1 s L 1 st = r 2 [ r 3 F 32 F 13 F 12 ( r 3 F 33 1 ) ] r 3 [ r 2 F 12 F 23 F 13 ( r 2 F 22 1 ) ] = r 2 r 3 r 2 = r 3 L 1 s L 1 st
r 2 [ ( 1 r 3 F 33 ) ( 1 r 1 F 11 ) r 1 r 3 F 31 F 13 ] r 3 [ ( 1 r 2 F 22 ) ( 1 r 1 F 11 ) r 1 r 2 F 21 F 12 ] = r 2 ( 1 r 3 A 2 A r 1 A 1 A ) r 3 ( 1 r 2 A 2 A r 1 A 1 A ) r 3 L 1 s L 1 st ( 1 r 1 A 1 A ) 1 r 1 A 1 A r 3 A 2 A ( 1 L 1 s L 1 st )
L 1 s L 1 st = r 2 r 1 ( 1 + r 2 A 2 A ) r 2 = r 1 L 1 s L 1 st 1 L 1 s L 1 st r 1 A 2 A
r 2 [ 1 r 1 A 1 A r 1 A 2 A ] r 2 [ 1 r 1 A 1 A r 1 r 2 A 2 A A 1 A ] r 1 L 1 s L 1 st ( 1 r 1 A 1 A ) 1 r 1 A 1 A r 1 A 2 A ( 1 r 1 A 1 A L 1 s L 1 st )
L 1 s L 1 st = r 2 ( 1 + r 3 A 2 A ) r 3 ( 1 + r 2 A 2 A ) r 2 = r 3 L 1 s L 1 st 1 + r 3 A 2 A ( 1 L 1 s L 1 st )
r 2 [ 1 r 1 A 1 A r 1 r 3 A 1 A A 2 A ] r 3 [ 1 r 1 A 1 A r 1 r 2 A 1 A A 2 A ] r 3 L 1 s L 1 st ( 1 r 1 A 1 A ) 1 r 1 A 1 A r 1 r 3 A 1 A A 2 A ( 1 L 1 s L 1 st )
L 1 s L 1 st = r 2 ( 1 + r 3 A 2 c A A 2 c ) r 3 ( 1 + r 2 A 2 c A A 2 c ) r 2 = r 3 L 1 s L 1 st 1 + r 3 A 2 c A A 2 c ( 1 L 1 s L 1 st )
r 2 [ 1 r 1 A 1 A r 1 r 3 A 2 c A A 1 A A 2 c ] r 3 [ 1 r 1 A 1 A r 1 r 2 A 2 c A A 1 A A 2 c ] r 3 L 1 s L 1 st ( 1 r 1 A 1 A ) 1 r 1 A 1 A r 1 r 3 A 1 A A 2 c A A 2 c ( 1 L 1 s L 1 st )
A 1 A
A 2 A
A 1 A
A 2 c A A 2 c
A 1 A
A 2 A
A 1 A
A 2 c A A 2 c
A 2 c A A 2 c
A 1 A
A 2 c A A 2 c