Abstract

Equations have been derived that allow differences in reflectometer readings of the same sample with various reflectometers to be minimized by placing all reflectances on an absolute basis. For pigmented vehicle films the Fresnel reflectances and transmittances at the air–vehicle interface can be calculated for an arbitrary incident angular light distribution. The form of the reflected and transmitted angular distributions can also be calculated. The equations have been used to determine absolute reflectances of titanium dioxide pigmented films from meter readings of two reflectometers—the General Electric Recording and the Colormaster.

© 1971 Optical Society of America

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References

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  1. A. M. Weinberg, E. P. Wigner, The Physical Theory of Neutron Chain Reactors (U. of Chicago Press, Chicago, 1958), Chaps. 8 and 9.
  2. H. J. McNicholas, J. Res. Nat. Bur. Stand. (U.S.) 1, 29 (1928).
  3. W. M. Wendlandt, Modern Aspects of Reflectance Spectroscopy (Plenum, New York, 1968), pp. 143–169.
  4. G. Wyszecki, W. S. Stiles, Color Science—Concepts and Methods, Quantitative Data and Formulas (Wiley, New York, 1967), pp. 198–201.
  5. J. A. Jacquez, H. L. Kuppenheim, J. Opt. Soc. Amer. 45, 460 (1955).
    [CrossRef]
  6. M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964).
  7. D. B. Judd, J. Res. Nat. Bur. Stand. (U.S.) 29, 329 (1942).
  8. R. G. Giovanelli, Opt. Acta 3, 127 (1956).
    [CrossRef]
  9. R. G. Giovanelli, Opt. Acta 2, 153 (1955).
    [CrossRef]
  10. J. L. Saunderson, J. Opt. Soc. Amer. 32, 727 (1942).
    [CrossRef]
  11. U. Zorll, Deut. Farben Z. 17, 6 (1963).
  12. W. E. K. Middleton, C. L. Saunders, J. Opt. Soc. Amer. 41, 419 (1951).
    [CrossRef]
  13. G. W. Gordon-Smith, Proc. Phys. Soc. (London) B65, 275 (1952).
  14. J. S. Laufer, J. Opt. Soc. Amer. 49, 1135 (1959).
  15. A. C. Hardy, J. Opt. Soc. Amer. 28, 360 (1938).
    [CrossRef]
  16. L. S. White, unpublished data.
  17. L. G. Glasser, D. J. Troy, J. Opt. Soc. Amer. 42, 652 (1952).
    [CrossRef]
  18. J. H. Perry, Chemical Engineers Handbook (McGraw-Hill, New York, 1968), pp. 10–37 to 10–39.

1963 (1)

U. Zorll, Deut. Farben Z. 17, 6 (1963).

1959 (1)

J. S. Laufer, J. Opt. Soc. Amer. 49, 1135 (1959).

1956 (1)

R. G. Giovanelli, Opt. Acta 3, 127 (1956).
[CrossRef]

1955 (2)

R. G. Giovanelli, Opt. Acta 2, 153 (1955).
[CrossRef]

J. A. Jacquez, H. L. Kuppenheim, J. Opt. Soc. Amer. 45, 460 (1955).
[CrossRef]

1952 (2)

G. W. Gordon-Smith, Proc. Phys. Soc. (London) B65, 275 (1952).

L. G. Glasser, D. J. Troy, J. Opt. Soc. Amer. 42, 652 (1952).
[CrossRef]

1951 (1)

W. E. K. Middleton, C. L. Saunders, J. Opt. Soc. Amer. 41, 419 (1951).
[CrossRef]

1942 (2)

D. B. Judd, J. Res. Nat. Bur. Stand. (U.S.) 29, 329 (1942).

J. L. Saunderson, J. Opt. Soc. Amer. 32, 727 (1942).
[CrossRef]

1938 (1)

A. C. Hardy, J. Opt. Soc. Amer. 28, 360 (1938).
[CrossRef]

1928 (1)

H. J. McNicholas, J. Res. Nat. Bur. Stand. (U.S.) 1, 29 (1928).

Born, M.

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964).

Giovanelli, R. G.

R. G. Giovanelli, Opt. Acta 3, 127 (1956).
[CrossRef]

R. G. Giovanelli, Opt. Acta 2, 153 (1955).
[CrossRef]

Glasser, L. G.

L. G. Glasser, D. J. Troy, J. Opt. Soc. Amer. 42, 652 (1952).
[CrossRef]

Gordon-Smith, G. W.

G. W. Gordon-Smith, Proc. Phys. Soc. (London) B65, 275 (1952).

Hardy, A. C.

A. C. Hardy, J. Opt. Soc. Amer. 28, 360 (1938).
[CrossRef]

Jacquez, J. A.

J. A. Jacquez, H. L. Kuppenheim, J. Opt. Soc. Amer. 45, 460 (1955).
[CrossRef]

Judd, D. B.

D. B. Judd, J. Res. Nat. Bur. Stand. (U.S.) 29, 329 (1942).

Kuppenheim, H. L.

J. A. Jacquez, H. L. Kuppenheim, J. Opt. Soc. Amer. 45, 460 (1955).
[CrossRef]

Laufer, J. S.

J. S. Laufer, J. Opt. Soc. Amer. 49, 1135 (1959).

McNicholas, H. J.

H. J. McNicholas, J. Res. Nat. Bur. Stand. (U.S.) 1, 29 (1928).

Middleton, W. E. K.

W. E. K. Middleton, C. L. Saunders, J. Opt. Soc. Amer. 41, 419 (1951).
[CrossRef]

Perry, J. H.

J. H. Perry, Chemical Engineers Handbook (McGraw-Hill, New York, 1968), pp. 10–37 to 10–39.

Saunders, C. L.

W. E. K. Middleton, C. L. Saunders, J. Opt. Soc. Amer. 41, 419 (1951).
[CrossRef]

Saunderson, J. L.

J. L. Saunderson, J. Opt. Soc. Amer. 32, 727 (1942).
[CrossRef]

Stiles, W. S.

G. Wyszecki, W. S. Stiles, Color Science—Concepts and Methods, Quantitative Data and Formulas (Wiley, New York, 1967), pp. 198–201.

Troy, D. J.

L. G. Glasser, D. J. Troy, J. Opt. Soc. Amer. 42, 652 (1952).
[CrossRef]

Weinberg, A. M.

A. M. Weinberg, E. P. Wigner, The Physical Theory of Neutron Chain Reactors (U. of Chicago Press, Chicago, 1958), Chaps. 8 and 9.

Wendlandt, W. M.

W. M. Wendlandt, Modern Aspects of Reflectance Spectroscopy (Plenum, New York, 1968), pp. 143–169.

White, L. S.

L. S. White, unpublished data.

Wigner, E. P.

A. M. Weinberg, E. P. Wigner, The Physical Theory of Neutron Chain Reactors (U. of Chicago Press, Chicago, 1958), Chaps. 8 and 9.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964).

Wyszecki, G.

G. Wyszecki, W. S. Stiles, Color Science—Concepts and Methods, Quantitative Data and Formulas (Wiley, New York, 1967), pp. 198–201.

Zorll, U.

U. Zorll, Deut. Farben Z. 17, 6 (1963).

Deut. Farben Z. (1)

U. Zorll, Deut. Farben Z. 17, 6 (1963).

J. Opt. Soc. Amer. (6)

W. E. K. Middleton, C. L. Saunders, J. Opt. Soc. Amer. 41, 419 (1951).
[CrossRef]

J. S. Laufer, J. Opt. Soc. Amer. 49, 1135 (1959).

A. C. Hardy, J. Opt. Soc. Amer. 28, 360 (1938).
[CrossRef]

L. G. Glasser, D. J. Troy, J. Opt. Soc. Amer. 42, 652 (1952).
[CrossRef]

J. A. Jacquez, H. L. Kuppenheim, J. Opt. Soc. Amer. 45, 460 (1955).
[CrossRef]

J. L. Saunderson, J. Opt. Soc. Amer. 32, 727 (1942).
[CrossRef]

J. Res. Nat. Bur. Stand. (U.S.) (2)

D. B. Judd, J. Res. Nat. Bur. Stand. (U.S.) 29, 329 (1942).

H. J. McNicholas, J. Res. Nat. Bur. Stand. (U.S.) 1, 29 (1928).

Opt. Acta (2)

R. G. Giovanelli, Opt. Acta 3, 127 (1956).
[CrossRef]

R. G. Giovanelli, Opt. Acta 2, 153 (1955).
[CrossRef]

Proc. Phys. Soc. (London) (1)

G. W. Gordon-Smith, Proc. Phys. Soc. (London) B65, 275 (1952).

Other (6)

L. S. White, unpublished data.

J. H. Perry, Chemical Engineers Handbook (McGraw-Hill, New York, 1968), pp. 10–37 to 10–39.

A. M. Weinberg, E. P. Wigner, The Physical Theory of Neutron Chain Reactors (U. of Chicago Press, Chicago, 1958), Chaps. 8 and 9.

W. M. Wendlandt, Modern Aspects of Reflectance Spectroscopy (Plenum, New York, 1968), pp. 143–169.

G. Wyszecki, W. S. Stiles, Color Science—Concepts and Methods, Quantitative Data and Formulas (Wiley, New York, 1967), pp. 198–201.

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964).

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Figures (5)

Fig. 1
Fig. 1

Current distribution functions of uniformly diffused, externally incident light.

Fig. 2
Fig. 2

Current distribution functions of uniformly diffused, internally incident light.

Fig. 3
Fig. 3

Angular distribution functions for uniformly diffused, incident light.

Fig. 4
Fig. 4

Goniophotometric curves of glossy TiO2 paint film and smoked MgO.

Fig. 5
Fig. 5

Simplified optical system of Colormaster differential calorimeter.

Tables (4)

Tables Icon

Table I Current Distribution Functions for Uniformly Diffused, Externally Incident Light (n2/n1 = 1.50000)

Tables Icon

Table II Current Distribution Functions of Uniformly Diffused, Internally Incident Light (n2/n1 = 1/1.50000)

Tables Icon

Table III Absolute Reflectance of Fumed MgO

Tables Icon

Table IV Comparison of Reflectometers

Equations (47)

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d I = f ( Ω ) d Ω ,
d J = f ( Ω ) cos θ d Ω .
R j k ( m ) = 2 π f j k ( m ) ( Ω r ) cos θ r d Ω r 2 π f ( m ) ( Ω i ) cos θ i d Ω i ,
R s a ( a ) = 2 π f s a ( a ) ( Ω r ) cos θ r d Ω r 2 π f ( a ) ( Ω i ) cos θ i d Ω i ,
R p v ( a ) = 2 π f p v ( a ) ( Ω r ) cos θ r d Ω r 2 π f ( a ) ( Ω i ) cos θ i d Ω i .
R s a ( a d ) = f s a ( a d ) / f ( a d ) ,
R p v ( a d ) = f p v ( a d ) / f ( a d ) ,
d J ( θ ) / J T F ( θ ) d θ = sin 2 θ d θ ,
M j k = α Ω r f j k ( a ) ( Ω r ) cos θ r d Ω r ,
M p v M s a = Ω r f p v ( a ) ( Ω r ) cos θ r d Ω r Ω r f s a ( a ) ( Ω r ) cos θ r d Ω r .
β j k = Ω r f j k ( a ) ( Ω r ) cos θ r d Ω r 2 π f j k ( a ) ( Ω r ) cos θ r d Ω r ,
R p v ( a ) = [ ( β s a M p v ) / ( β p v M s a ) ] R s a ( a ) .
R p v ( a ) = ( M p v / M s a ) R s a ( a ) β s a / β p v = 1
d R ( F ) = [ f ( ) ( Ω r ) + f ( ) ( Ω r ) ] cos θ r d Ω r 2 π f ( Ω i ) cos θ i d Ω i ,
d R ( F ) ( θ r ) = [ f ( ) ( Ω r ) + f ( ) ( Ω r ) ] sin 2 θ r d θ r 0 π / 2 f ( Ω i ) sin 2 θ i d θ i ,
f ( Ω ) = 1 2 π 0 2 π f ( Ω ) d φ
d ( ) J ( θ r ) / d ( ) J ( θ i ) = sin 2 ( θ i - θ t ) / sin 2 ( θ i + θ t )
d ( ) J ( θ r ) / d ( ) J ( θ i ) = tan 2 ( θ i - θ t ) / tan 2 ( θ i + θ i ) .
d ( ) J ( θ r ) / d ( ) J ( θ i ) = f ( ) ( Ω r ) / f ( ) ( Ω i )
d ( ) J ( θ r ) / d ( ) J ( θ i ) = f ( ) ( Ω r ) / ( ) f ( Ω i ) .
f ( ) ( Ω i ) = f ( ) ( Ω i ) = 1 2 f ( Ω i ) .
R ( F ) = 1 2 0 π / 2 f ( Ω i ) [ sin 2 ( θ i - θ t ) sin 2 ( θ i + θ t ) + tan 2 ( θ i - θ t ) tan 2 ( θ i + θ t ) ] sin 2 θ i d θ i 0 π / 2 f ( Ω i ) sin 2 θ i d θ i
T ( F ) = 1 2 0 π / 2 f ( Ω i ) [ sin 2 θ i sin 2 θ t sin 2 ( θ i + θ t ) + sin 2 θ i sin 2 θ t sin 2 ( θ i + θ t ) cos 2 ( θ i - θ t ) ] sin 2 θ i d θ i 0 π / 2 f ( Ω i ) sin 2 θ i d θ i .
R ( F ) + T ( F ) = 1.
d J ( θ i ) / J T i F ( θ i ) d θ i ,
f ( θ i ) = ( J T / π sin 2 θ i ) F ( θ i ) .
d J ( θ r ) J T i F ( θ r ) d θ r = 1 2 [ sin 2 ( θ i - θ t ) sin 2 ( θ i + θ t ) + tan 2 ( θ i - θ t ) tan 2 ( θ i + θ t ) ] F ( θ i ) d θ r ,
d J ( θ t ) J T i F ( θ t ) d θ t = 1 2 [ sin 2 θ i sin 2 θ t sin 2 ( θ i + θ t ) + sin 2 θ i sin 2 θ t sin 2 ( θ i + θ t ) cos 2 ( θ i - θ t ) ] tan θ i tan θ t F ( θ i ) d θ t .
F ( θ t ) = ( tan θ i / tan θ t ) [ F ( θ i ) - F ( θ r ) ] .
R ( F d ) = 0 π / 2 1 2 [ sin 2 ( θ i - θ t ) sin 2 ( θ i + θ t ) + tan 2 ( θ i - θ t ) tan 2 ( θ i + θ t ) ] sin 2 θ r d θ r .
R p v ( v ) = 2 π f p v ( v ) ( Ω r ) cos θ r d θ r T e ( F ) 2 π f ( a ) ( Ω i ) cos θ i d Ω i + R i ( F ) 2 π f p v ( v ) ( Ω r ) cos θ r d Ω r ,
R p v ( a ) = T i ( F ) 2 π f p v ( v ) ( Ω r ) cos θ r d Ω r + R e ( F ) 2 π f ( a ) ( Ω i ) cos θ i d Ω i 2 π f ( a ) ( Ω i ) cos θ i d Ω i ,
2 π f p v g v ) ( Ω r ) cos θ r d Ω r
R p v ( v ) = R p v ( a ) - R e ( F ) T e ( F ) T i ( F ) + R i ( F ) [ R p v ( a ) - R e ( F ) ] .
R p a ( a ) = ( M p a / 100 ) R s a ( a ) ,
R e ( F ) = 0 π / 2 1 2 [ sin 2 ( θ i - θ t ) sin 2 ( θ i + θ t ) + tan 2 ( θ i - θ t ) tan 2 ( θ i + θ t ) ] δ ( θ i - 6 ° ) d θ i ,
R p v ( a ) = ( M p v / 100 ) R s a ( a ) ,
R p v ( v ) = R p v ( a ) - 0.0400 0.3861 + 05978 ( R p v ( a ) - 0.0400 ) .
R p a ( a ) = ( M p a / 100 ) R s a ( a ) ,
M j k = α r i = 1 N Ω r i D ( Ω r ) f j k ( a ) ( Ω r ) cos θ r d Ω r .
β p v = Ω r f p v ( a ) ( Ω r ) cos θ r d Ω r 2 π f ( D ) p v ( Ω r ) ( a ) cos θ r d Ω r + R e ( F ) Ω i f ( a ) ( Ω i ) cos θ i d Ω i
β s a β p v = Ω r f s a ( a ) ( Ω r ) cos θ r d Ω r 2 π f ( D ) p v ( Ω r ) ( a ) cos θ r d Ω r 2 π f s a ( a ) ( Ω r ) cos θ r d Ω r Ω r f p v ( a ) ( Ω r ) cos θ d Ω r + R e ( F ) R s a ( a ) M s a M p v .
β s a / β p v = 0.979 + ( 5.15 / M p v ) .
R p v ( a ) = [ 0.979 + ( 5.15 / M p v ) ] ( M p v / 100 ) R s a ( a ) ,
R p v ( v ) = R p v ( a ) - 0.0503 0.3820 + 0.5978 [ R p v ( a ) - 0.0503 ] .
β s a π 0 6 ° 4 3 f s a ( a ) sin 2 θ r d θ r π 0 70 ° f s a ( a ) sin 2 θ r d θ r + π 70 ° 90 ° ( 2.0 / 3.0 ) f s a ( a ) sin 2 θ r d θ r = 0.0142.
Ω r f p v ( a ) ( Ω r ) cos θ r d Ω r / 2 π f p v ( a ) ( Ω r ) cos θ r d Ω r π 0 3 ° 4 3 f p v sin 2 θ r d θ r π 0 70 ° f p v ( a ) sin 2 θ r d θ r + π 70 ° 90 ° ( 0.4 / 0.8 ) f p v ( a ) sin 2 θ r d θ r = 0.0145.

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