Abstract

Concepts, terminology, and symbols are presented for specifying and relating directional variations in reflectance and emissivity of an opaque surface element. Their relationship to more familiar concepts, including those of perfectly diffuse and specular reflectance, is given, and they are applied to illustrative examples. It is shown that, when the usual reciprocity relationship holds, the reflectance for a ray incident on an opaque surface element is related by Kirchhoff's law to the emissivity of that element for a ray emitted along the same line in the opposite sense.

© 1965 Optical Society of America

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References

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  1. C. von Fragstein, Optik 12, 60 (1955).
  2. F. E. Nicodemus, Am. J. Phys. 31, 368 (1963).
    [CrossRef]
  3. G. Kelton et al., Infrared Phys. 3, 139 (1963).
    [CrossRef]
  4. E. E. Bell, Proc. Inst. Radio Engrs. 47, 1432 (1959).
  5. R. C. Jones, Appl. Opt. 1, 607 (1962).
    [CrossRef]
  6. J. C. DeVos, Physica 20, 669 (1954).
    [CrossRef]
  7. D. K. Edwards, J. T. Gier, K. E. Nelson, R. D. Roddick, J. Opt. Soc. Am. 51, 1279 (1961).
    [CrossRef]
  8. D. E. Kerr, “Application of the Lorentz Reciprocity Theorem to Scattering”, Appendix A in Propagation of Short Radio Waves (McGraw-Hill, New York, 1951), 1st ed., Vol. 13 of MIT Radiation Laboratory Series, p. 693.
  9. A. T. DeHoop, Appl. Sci. Res. Sec. B 8, 135 (1960).
    [CrossRef]
  10. A. Sommerfeld, Jahrbuch d. drahtl. Telegraphie 37–38, 167 (1931).
  11. D. S. Saxon, Phys. Rev. 100, 1771 (1955).
    [CrossRef]
  12. H. J. McNicholas, J. Res. Natl. Bur. Std. 1, 29 (1928).
    [CrossRef]
  13. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 380.
  14. Committee on Colorimetry, Optical Society of America, The Science of Color (Thomas Y. Crowell Co., New York, 1953), p. 178.
  15. E. T. de la Perrelle, T. S. Moss, H. Herbert, Infrared Phys. 3, 35 (1963).
    [CrossRef]
  16. M. Planck, Theory of Heat (Macmillan, New York, 1957), transl. by H. L. Brose, Vol. V of Introduction to Theoretical Physics, p. 193.
  17. C. von Fragstein, Optica Acta 2, 16 (1955).
    [CrossRef]
  18. H. von Helmholtz, Helmholtz's Treatise on Physiological Optics (Optical Society of America, Washington, D.C., 1924) transl. from third German ed., J. P. C. Southall, ed., Vol. 1, p. 231.
  19. B. W. Hapke, H. Van Horn, J. Geophys. Res. 68, 4545 (1963).
    [CrossRef]
  20. B. W. Hapke, J. Geophys. Res. 68, 4571 (1963).
    [CrossRef]
  21. V. Twersky, IRE Trans. Antennas and Propagation AP-5, 81 (1957).
    [CrossRef]
  22. V. Twersky, J. Res. Natl. Bur. Std. 64D, 715 (1960).
  23. V. Twersky, J. Opt. Soc. Am. 52, 145 (1962).
    [CrossRef]
  24. H. E. Bennett, J. O. Porteus, J. Opt. Soc. Am. 51, 123 (1961).
    [CrossRef]
  25. R. K. McDonald, in Proc. Infrared Information Symposia, 5, no. 3, p. 153 (1960).
  26. H. H. Blau, J. L. Miles, L. E. Ashman, Thermal Radiation Characteristics of Solid Materials—A Review, (Arthur D. Little, Inc., Cambridge, Mass., 1958, Scientific Report No. 1, AFCRC-TN-58-132).
  27. G. Bauer, Optik 18, 603 (1961).
  28. D. K. Edwards, “Measurement of Thermal Radiation Characteristics”, Inst. Environmental Sciences Proc., 1963.

1963 (6)

E. T. de la Perrelle, T. S. Moss, H. Herbert, Infrared Phys. 3, 35 (1963).
[CrossRef]

B. W. Hapke, H. Van Horn, J. Geophys. Res. 68, 4545 (1963).
[CrossRef]

B. W. Hapke, J. Geophys. Res. 68, 4571 (1963).
[CrossRef]

F. E. Nicodemus, Am. J. Phys. 31, 368 (1963).
[CrossRef]

G. Kelton et al., Infrared Phys. 3, 139 (1963).
[CrossRef]

D. K. Edwards, “Measurement of Thermal Radiation Characteristics”, Inst. Environmental Sciences Proc., 1963.

1962 (2)

1961 (3)

1960 (3)

R. K. McDonald, in Proc. Infrared Information Symposia, 5, no. 3, p. 153 (1960).

V. Twersky, J. Res. Natl. Bur. Std. 64D, 715 (1960).

A. T. DeHoop, Appl. Sci. Res. Sec. B 8, 135 (1960).
[CrossRef]

1959 (1)

E. E. Bell, Proc. Inst. Radio Engrs. 47, 1432 (1959).

1957 (1)

V. Twersky, IRE Trans. Antennas and Propagation AP-5, 81 (1957).
[CrossRef]

1955 (3)

C. von Fragstein, Optica Acta 2, 16 (1955).
[CrossRef]

D. S. Saxon, Phys. Rev. 100, 1771 (1955).
[CrossRef]

C. von Fragstein, Optik 12, 60 (1955).

1954 (1)

J. C. DeVos, Physica 20, 669 (1954).
[CrossRef]

1931 (1)

A. Sommerfeld, Jahrbuch d. drahtl. Telegraphie 37–38, 167 (1931).

1928 (1)

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

Ashman, L. E.

H. H. Blau, J. L. Miles, L. E. Ashman, Thermal Radiation Characteristics of Solid Materials—A Review, (Arthur D. Little, Inc., Cambridge, Mass., 1958, Scientific Report No. 1, AFCRC-TN-58-132).

Bauer, G.

G. Bauer, Optik 18, 603 (1961).

Bell, E. E.

E. E. Bell, Proc. Inst. Radio Engrs. 47, 1432 (1959).

Bennett, H. E.

Blau, H. H.

H. H. Blau, J. L. Miles, L. E. Ashman, Thermal Radiation Characteristics of Solid Materials—A Review, (Arthur D. Little, Inc., Cambridge, Mass., 1958, Scientific Report No. 1, AFCRC-TN-58-132).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 380.

de la Perrelle, E. T.

E. T. de la Perrelle, T. S. Moss, H. Herbert, Infrared Phys. 3, 35 (1963).
[CrossRef]

DeHoop, A. T.

A. T. DeHoop, Appl. Sci. Res. Sec. B 8, 135 (1960).
[CrossRef]

DeVos, J. C.

J. C. DeVos, Physica 20, 669 (1954).
[CrossRef]

Edwards, D. K.

D. K. Edwards, “Measurement of Thermal Radiation Characteristics”, Inst. Environmental Sciences Proc., 1963.

D. K. Edwards, J. T. Gier, K. E. Nelson, R. D. Roddick, J. Opt. Soc. Am. 51, 1279 (1961).
[CrossRef]

Gier, J. T.

Hapke, B. W.

B. W. Hapke, H. Van Horn, J. Geophys. Res. 68, 4545 (1963).
[CrossRef]

B. W. Hapke, J. Geophys. Res. 68, 4571 (1963).
[CrossRef]

Herbert, H.

E. T. de la Perrelle, T. S. Moss, H. Herbert, Infrared Phys. 3, 35 (1963).
[CrossRef]

Jones, R. C.

Kelton, G.

G. Kelton et al., Infrared Phys. 3, 139 (1963).
[CrossRef]

Kerr, D. E.

D. E. Kerr, “Application of the Lorentz Reciprocity Theorem to Scattering”, Appendix A in Propagation of Short Radio Waves (McGraw-Hill, New York, 1951), 1st ed., Vol. 13 of MIT Radiation Laboratory Series, p. 693.

McDonald, R. K.

R. K. McDonald, in Proc. Infrared Information Symposia, 5, no. 3, p. 153 (1960).

McNicholas, H. J.

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

Miles, J. L.

H. H. Blau, J. L. Miles, L. E. Ashman, Thermal Radiation Characteristics of Solid Materials—A Review, (Arthur D. Little, Inc., Cambridge, Mass., 1958, Scientific Report No. 1, AFCRC-TN-58-132).

Moss, T. S.

E. T. de la Perrelle, T. S. Moss, H. Herbert, Infrared Phys. 3, 35 (1963).
[CrossRef]

Nelson, K. E.

Nicodemus, F. E.

F. E. Nicodemus, Am. J. Phys. 31, 368 (1963).
[CrossRef]

Planck, M.

M. Planck, Theory of Heat (Macmillan, New York, 1957), transl. by H. L. Brose, Vol. V of Introduction to Theoretical Physics, p. 193.

Porteus, J. O.

Roddick, R. D.

Saxon, D. S.

D. S. Saxon, Phys. Rev. 100, 1771 (1955).
[CrossRef]

Sommerfeld, A.

A. Sommerfeld, Jahrbuch d. drahtl. Telegraphie 37–38, 167 (1931).

Twersky, V.

V. Twersky, J. Opt. Soc. Am. 52, 145 (1962).
[CrossRef]

V. Twersky, J. Res. Natl. Bur. Std. 64D, 715 (1960).

V. Twersky, IRE Trans. Antennas and Propagation AP-5, 81 (1957).
[CrossRef]

Van Horn, H.

B. W. Hapke, H. Van Horn, J. Geophys. Res. 68, 4545 (1963).
[CrossRef]

von Fragstein, C.

C. von Fragstein, Optica Acta 2, 16 (1955).
[CrossRef]

C. von Fragstein, Optik 12, 60 (1955).

von Helmholtz, H.

H. von Helmholtz, Helmholtz's Treatise on Physiological Optics (Optical Society of America, Washington, D.C., 1924) transl. from third German ed., J. P. C. Southall, ed., Vol. 1, p. 231.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 380.

Am. J. Phys. (1)

F. E. Nicodemus, Am. J. Phys. 31, 368 (1963).
[CrossRef]

Appl. Opt. (1)

Appl. Sci. Res. Sec. B (1)

A. T. DeHoop, Appl. Sci. Res. Sec. B 8, 135 (1960).
[CrossRef]

Infrared Phys. (2)

E. T. de la Perrelle, T. S. Moss, H. Herbert, Infrared Phys. 3, 35 (1963).
[CrossRef]

G. Kelton et al., Infrared Phys. 3, 139 (1963).
[CrossRef]

Inst. Environmental Sciences Proc. (1)

D. K. Edwards, “Measurement of Thermal Radiation Characteristics”, Inst. Environmental Sciences Proc., 1963.

IRE Trans. Antennas and Propagation (1)

V. Twersky, IRE Trans. Antennas and Propagation AP-5, 81 (1957).
[CrossRef]

J. Geophys. Res. (2)

B. W. Hapke, H. Van Horn, J. Geophys. Res. 68, 4545 (1963).
[CrossRef]

B. W. Hapke, J. Geophys. Res. 68, 4571 (1963).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Res. Natl. Bur. Std. (2)

V. Twersky, J. Res. Natl. Bur. Std. 64D, 715 (1960).

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

Jahrbuch d. drahtl. Telegraphie (1)

A. Sommerfeld, Jahrbuch d. drahtl. Telegraphie 37–38, 167 (1931).

Optica Acta (1)

C. von Fragstein, Optica Acta 2, 16 (1955).
[CrossRef]

Optik (2)

C. von Fragstein, Optik 12, 60 (1955).

G. Bauer, Optik 18, 603 (1961).

Phys. Rev. (1)

D. S. Saxon, Phys. Rev. 100, 1771 (1955).
[CrossRef]

Physica (1)

J. C. DeVos, Physica 20, 669 (1954).
[CrossRef]

Proc. Infrared Information Symposia (1)

R. K. McDonald, in Proc. Infrared Information Symposia, 5, no. 3, p. 153 (1960).

Proc. Inst. Radio Engrs. (1)

E. E. Bell, Proc. Inst. Radio Engrs. 47, 1432 (1959).

Other (6)

H. H. Blau, J. L. Miles, L. E. Ashman, Thermal Radiation Characteristics of Solid Materials—A Review, (Arthur D. Little, Inc., Cambridge, Mass., 1958, Scientific Report No. 1, AFCRC-TN-58-132).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 380.

Committee on Colorimetry, Optical Society of America, The Science of Color (Thomas Y. Crowell Co., New York, 1953), p. 178.

H. von Helmholtz, Helmholtz's Treatise on Physiological Optics (Optical Society of America, Washington, D.C., 1924) transl. from third German ed., J. P. C. Southall, ed., Vol. 1, p. 231.

M. Planck, Theory of Heat (Macmillan, New York, 1957), transl. by H. L. Brose, Vol. V of Introduction to Theoretical Physics, p. 193.

D. E. Kerr, “Application of the Lorentz Reciprocity Theorem to Scattering”, Appendix A in Propagation of Short Radio Waves (McGraw-Hill, New York, 1951), 1st ed., Vol. 13 of MIT Radiation Laboratory Series, p. 693.

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Figures (1)

Fig. 1
Fig. 1

Geometry of incident and reflected elementary beams. (Z axis is chosen along the normal to the surface element at 0.)

Tables (1)

Tables Icon

Table I Radiometric Quantities, Symbols, Definations, and Unitsa

Equations (44)

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N 2 P cos θ A Ω [ W · cm 2 · sr 1 ] ,
d P i = N i cos θ i d A d Ω i = N i d A d Ω i [ W ] ,
d P r = d N r d A d Ω r [ W ] .
ρ = d P r / d P i = d N r d Ω r / N i d Ω i [ dimensionless ] .
ρ / d Ω r = d N r / N i d Ω i [ sr 1 ] .
d P i = N i d A d Ω i [ W ] ,
d P r = d N r d A d Ω r [ W ] .
d ρ = d P r / d P i = d N r d Ω r / N i d Ω i = ρ ( θ i , φ i , θ r , φ r ) d Ω r [ dimensionless ] ,
ρ ( θ i , φ i , θ r , φ r ) d N r / N i d Ω i = d N r / d H i [ sr 1 ] .
ρ ( θ 1 , φ 1 , θ 2 , φ 2 ) = ρ ( θ 2 , φ 2 , θ 1 , φ 1 ) [ sr 1 ] .
N r ( θ r , φ r ) = h ρ ( θ i , φ i , θ r , φ r ) N i ( θ i , φ i ) d Ω i [ W · cm 2 · sr 1 ] ,
h f ( θ , φ ) d Ω 0 2 π 0 π / 2 f ( θ , φ ) sin θ d θ d φ , h f ( θ , φ ) d Ω 0 2 π 0 π / 2 f ( θ , φ ) sin θ cos θ d θ d φ .
ρ d P r / d P i [ dimensionless ] ,
d P i = d H i ( θ i , φ i ) d A [ W ] .
d N r ( θ r , φ r ) = ρ ( θ i , φ i , θ r , φ r ) d H i ( θ i , φ i ) = ρ ( θ i , φ i , θ r , φ r ) d P i / d A [ W · cm 2 · sr 1 ] .
d P r = d A h d N r ( θ r , φ r ) d Ω r = d P i h ρ ( θ i , φ i , θ r , φ r ) d Ω r = d P i ρ d i ( θ i , φ i ) [ W ] ,
ρ d i ( θ i , φ i ) h ρ ( θ i , φ i , θ r , φ r ) d Ω r [ dimensionless ] .
N r ( θ r , φ r ) = N i h ρ ( θ i , φ i , θ r , φ r ) d Ω i = ρ d r ( θ r , φ r ) N i [ W · c m 2 · sr 1 ] ;
ρ d r ( θ r , φ r ) h ρ ( θ i , φ i , θ r , φ r ) d Ω i [ dimensionless ] .
ρ d i ( θ 1 , φ 1 ) = ρ d r ( θ 1 , φ 1 ) = ρ d ( θ 1 , φ 1 ) [ dimensionless ] .
N a + N r = d ( θ 1 , φ 1 ) N b ( T ) + ρ d r ( θ 1 , φ 1 ) N b ( T ) = N b ( T ) [ W · c m 2 · sr 1 ] .
N e + N ir = α d ( θ 1 , φ 1 ) N b ( T ) + ρ d i ( θ 1 , φ 1 ) N b ( T ) = N b ( T ) [ W · c m 2 · sr 1 ] .
d ( θ 1 , φ 1 ) = 1 ρ d r ( θ 1 , φ 1 ) = 1 ρ d i ( θ 1 , φ 1 ) = α d ( θ 1 , φ 1 ) [ dimensionless ] .
= 1 ρ = α [ dimensionless ] .
ρ = d P r / d P i = d A h N r d Ω d A h N i d Ω = N i h ρ d ( θ , φ ) d Ω N i h d Ω = 1 π h ρ d ( θ , φ ) d Ω [ dimensionless ]
ρ d = ρ h d Ω = π ρ [ dimensionless ] .
ρ = P r P i = h N r d Ω d A h N i d Ω d A = ρ d h N i d Ω d A h N i d Ω d A = ρ d = π ρ [ dimensionless ] ,
ρ = N r / H i = ρ / π [ sr 1 ] .
N r ( θ , φ ± π ) = ρ d ( θ , φ ) N i ( θ , φ ) [ W · c m 2 · sr 1 ] .
ρ ( θ i , φ i , θ r , φ r ) = 2 ρ d ( θ i , φ i ) δ ( sin 2 θ r sin 2 θ i ) δ ( φ r φ i ± π ) [ sr 1 ] ,
δ ( u ) = 0 for u 0 , δ ( u ) d u = 1 , and f ( u ) δ ( u ) d u = f ( 0 ) ,
P d = N r d Ω r d A = ( ρ / π ) N i Δ Ω i cos θ r d Ω r d A = ( ρ / π ) N i Δ Ω i Δ Ω r Δ A [ W ] .
P s = N r d Ω r d A = ρ d ( θ , φ ) N i Δ Ω r Δ A [ W ] .
P s P d = π Δ Ω i = π Δ Ω i cos θ i [ dimensionless ] .
u U V
P U t
J P Ω
W H } P A
N 2 P cos θ A Ω
P λ P λ
J λ J λ
W λ W λ
H λ H λ
N λ N λ

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