Abstract

The unidirectional reflectance of an imperfectly diffuse surface is defined in terms of the surface radiance and the equivalent hemispherical radiance of the source. Equations relating the unidirectional reflectance to the directional reflectance, the polar and azimuthal angles for incident and reflected energy, and the voltage signal of a phototube detector are derived. A reflectometer is described for measuring the monochromatic unidirectional reflectance over the visible region of the spectrum and data of a magnesium oxide coating, two white paints, and a dull and polished aluminum surface are presented.

© 1966 Optical Society of America

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Corrections

John. Levinson, "Errata: One-Stage Model for Visual Temporal Integration," J. Opt. Soc. Am. 56, 529_1-529 (1966)
https://www.osapublishing.org/josa/abstract.cfm?uri=josa-56-4-529_1

References

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  1. K. E. Torrance and E. M. Sparrow, J. Heat Transfer 87, 283 (1965).
    [Crossref]
  2. R. C. Birkebak and E. R. G. Eckert, J. Heat Transfer 87, 85 (1965).
    [Crossref]
  3. H. J. McNicholas, J. Res. Natl. Bur. Std,  1, 29 (1928).
    [Crossref]
  4. F. E. Nicodemus, Appl. Opt. 4, 767 (1965).
    [Crossref]
  5. D. K. Edwards, J. T. Gier, J. E. Nelson, and R. D. Roddick, J. Opt. Soc. Am. 51, 1279 (1961).
    [Crossref]
  6. J. T. Gier, R. V. Dunkle, and J. T. Bevans, J. Opt. Soc. Am. 44, 558 (1954).
    [Crossref]
  7. F. Mäder, Bull. Schweiz Electrotechn. Ver. 38, 632 (1947).
  8. J. A. Jacquez, W. McKeehan, J. Huss, J. M. Dimitroff, and H. F. Kuppenheim, J. Opt. Soc. Am. 45, 781 (1955).
    [Crossref]
  9. C. P. Tingwaldt, Optik 9, 323 (1952).
  10. W. E. K. Middleton and C. L. Sanders, Illuminating Eng. 48, 254 (1953).
  11. Dhetty Blet-Talbot, Rev. Opt. 34, 579 (1955).

1965 (3)

F. E. Nicodemus, Appl. Opt. 4, 767 (1965).
[Crossref]

K. E. Torrance and E. M. Sparrow, J. Heat Transfer 87, 283 (1965).
[Crossref]

R. C. Birkebak and E. R. G. Eckert, J. Heat Transfer 87, 85 (1965).
[Crossref]

1961 (1)

1955 (2)

1954 (1)

1953 (1)

W. E. K. Middleton and C. L. Sanders, Illuminating Eng. 48, 254 (1953).

1952 (1)

C. P. Tingwaldt, Optik 9, 323 (1952).

1947 (1)

F. Mäder, Bull. Schweiz Electrotechn. Ver. 38, 632 (1947).

1928 (1)

H. J. McNicholas, J. Res. Natl. Bur. Std,  1, 29 (1928).
[Crossref]

Bevans, J. T.

Birkebak, R. C.

R. C. Birkebak and E. R. G. Eckert, J. Heat Transfer 87, 85 (1965).
[Crossref]

Blet-Talbot, Dhetty

Dhetty Blet-Talbot, Rev. Opt. 34, 579 (1955).

Dimitroff, J. M.

Dunkle, R. V.

Eckert, E. R. G.

R. C. Birkebak and E. R. G. Eckert, J. Heat Transfer 87, 85 (1965).
[Crossref]

Edwards, D. K.

Gier, J. T.

Huss, J.

Jacquez, J. A.

Kuppenheim, H. F.

Mäder, F.

F. Mäder, Bull. Schweiz Electrotechn. Ver. 38, 632 (1947).

McKeehan, W.

McNicholas, H. J.

H. J. McNicholas, J. Res. Natl. Bur. Std,  1, 29 (1928).
[Crossref]

Middleton, W. E. K.

W. E. K. Middleton and C. L. Sanders, Illuminating Eng. 48, 254 (1953).

Nelson, J. E.

Nicodemus, F. E.

Roddick, R. D.

Sanders, C. L.

W. E. K. Middleton and C. L. Sanders, Illuminating Eng. 48, 254 (1953).

Sparrow, E. M.

K. E. Torrance and E. M. Sparrow, J. Heat Transfer 87, 283 (1965).
[Crossref]

Tingwaldt, C. P.

C. P. Tingwaldt, Optik 9, 323 (1952).

Torrance, K. E.

K. E. Torrance and E. M. Sparrow, J. Heat Transfer 87, 283 (1965).
[Crossref]

Appl. Opt. (1)

Bull. Schweiz Electrotechn. Ver. (1)

F. Mäder, Bull. Schweiz Electrotechn. Ver. 38, 632 (1947).

Illuminating Eng. (1)

W. E. K. Middleton and C. L. Sanders, Illuminating Eng. 48, 254 (1953).

J. Heat Transfer (2)

K. E. Torrance and E. M. Sparrow, J. Heat Transfer 87, 283 (1965).
[Crossref]

R. C. Birkebak and E. R. G. Eckert, J. Heat Transfer 87, 85 (1965).
[Crossref]

J. Opt. Soc. Am. (3)

J. Res. Natl. Bur. Std (1)

H. J. McNicholas, J. Res. Natl. Bur. Std,  1, 29 (1928).
[Crossref]

Optik (1)

C. P. Tingwaldt, Optik 9, 323 (1952).

Rev. Opt. (1)

Dhetty Blet-Talbot, Rev. Opt. 34, 579 (1955).

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Figures (11)

Fig. 1
Fig. 1

Beam and sample geometry.

Fig. 2
Fig. 2

Unidirectional reflectometer.

Fig. 3
Fig. 3

Angular divergence of detector beam as a function of source aperture.

Fig. 4
Fig. 4

Relative reflected flux γ of a 1-mm MgO coating as a function of reflected angle θ′ at λ=0.507 μ. (γ=1 when θ=θ′=θ0=θ0=15°, ϕ0=270°, ϕ′=ϕ0=90°); for a complete diffuser, γ=1 for all angles of incidence and reflection.

Fig. 5
Fig. 5

Relative reflected flux γ of a barium sulfate paint as a function of reflected angle θ′ at λ=0.507 μ (γ=1 when θ=θ′=θ0=θ0=15°, ϕ=270°, ϕ′=ϕ0=90°); for a complete diffuser, γ=1 for all angles of incidence and reflection.

Fig. 6
Fig. 6

Reflected energy from 1-mm MgO coating in the plane of incidence at λ=0.507 μ (incident energy polarized parallel to plane of incidence.

Fig. 7
Fig. 7

Unidirectional reflectance ρu of a zinc oxide paint versus polar angle of reflection θ′ at λ=0.533 μ.

Fig. 8
Fig. 8

Unidirectional reflectance ρu of a zinc oxide paint versus the azimuthal angle of reflection ϕ′ at λ=0.533 μ.

Fig. 9
Fig. 9

Directional reflectance ρd of a zinc oxide paint versus polar angle of reflection θ′ at λ=0.533 μ.

Fig. 10
Fig. 10

Maximum error Δρu/ρu in unidirectional reflectance of zinc oxide paint versus angle of reflection θ′ at λ=0.533 μ.

Fig. 11
Fig. 11

Relative unidirectional reflectance of dull and polished aluminum as a function of the reflected angle θ′ at λ=0.507 μ (γ=1 when θ0=30°, ϕ0=270°, θ0=30°, ϕ0=90°).

Tables (1)

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Table I Some reciprocal unidirectional reflectances of zinc oxide paint.

Equations (12)

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d F = B ( θ , ϕ ) cos θ d ω ,             ( W / cm 2 )
d F = B ( θ , ϕ ; θ ϕ ) cos θ d ω ,             ( W / cm 2 )
B ( θ , ϕ ) cos θ d ω B 0 cos θ d ω = π B 0 ,
ρ u ( θ , ϕ ; θ , ϕ ) π B ( θ , ϕ ; θ , ϕ ) / π B 0 = B ( θ , ϕ ; θ , ϕ ) / B 0 .
ρ u ( θ , ϕ ; θ , ϕ ) = ρ u ( θ , ϕ ; θ , ϕ ) .
ρ d ( θ , ϕ ) B ( θ , ϕ ; θ , ϕ ) cos θ d ω / π B 0 ,
ρ d ( θ , ϕ ) = ( 1 / π ) ρ u ( θ , ϕ ; θ , ϕ ) cos θ d ω .
d F d ω = d ω B cos θ d ω .
ρ u ( θ , ϕ ; θ , ϕ ) = π ρ d ( θ , ϕ ) d F ( θ , ϕ ; θ , ϕ ) cos θ d F ( θ , ϕ ; θ , ϕ ) d ω .
ρ u ( θ , ϕ ; θ , ϕ ) = π ρ d ( θ , ϕ ) V ( θ , ϕ ; θ , ϕ ) cos θ V ( θ , ϕ ; θ , ϕ ) d ω ,
ρ u ( θ i , ϕ j ; θ n , ϕ m ) = ρ d ( θ i , ϕ j ) V ( θ i , ϕ j ; θ n , ϕ m ) m = 1 M n = 1 N cos θ n sin θ n Δ θ n Δ ϕ m cos θ n m = 1 M n = 1 N V ( θ i , ϕ j ; θ n , ϕ m ) sin θ n Δ θ n Δ ϕ m ,
Δ ρ u ρ u = ± { [ 1 + 2 ( ρ u ρ d ) cos θ ] Δ V V + Δ ρ d ρ d } .