Abstract

Contrary to common belief, neither reciprocity of the bidirectional reflectance distribution function (BRDF) nor the directional form of Kirchhoff’s electromagnetic radiation law can be demonstrated on the basis of energy conservation. The BRDF is generally considered reciprocal as an extension of Helmholtz reciprocity, but Helmholtz reciprocity does not always hold. We describe the flaw in a thermodynamic demonstration of reciprocity that uses an enclosure calculation. Some conclusions can be drawn from the enclosure calculation, but reciprocity requires more restrictive conditions. We conclude that, although they can be violated, reciprocity and the directional form of Kirchhoff’s law generally hold because of the quantum-mechanical principle of time-reversal invariance, which applies to most materials.

© 1998 Optical Society of America

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References

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  1. R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer, 2nd ed. (Hemisphere, Washington, D.C., 1981), pp. 65–66.
  2. W. C. Snyder, Z. Wan, “Surface temperature correction for active infrared reflectance measurements of natural materials,” Appl. Opt. 35, 2216–2220 (1996).
    [CrossRef] [PubMed]
  3. W. C. Snyder, Z. Wan, “BRDF models to predict spectral reflectance and emissivity in the thermal infrared,” IEEE Trans. Geosci. Remote Sens. 36, 214–225 (1998).
    [CrossRef]
  4. F. Nicodemus, J. Richmond, J. Hsia, I. Ginsberg, T. Limperis, Geometrical Considerations and Nomenclature for Reflectance, NBS Monograph 160 (National Bureau of Standards, Washington, D.C., 1977), pp. 3–6.
  5. C. von Fragstein, “Über die Formulierung des Kirchhoffschen Gesetzes und ihre Bedeutung für eine zweckmässige Definition von Remissionszahlen,” Optik 12, 60–68 (1955).
  6. F. Clarke, D. Parry, “Helmholtz reciprocity: its validity and application to reflectometry,” Ltg. Res. Technol. 17, 1–11 (1985).
    [CrossRef]
  7. L. Levi, Applied Optics (Wiley, New York, 1968), p. 84.
  8. F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965), pp. 382–384.
  9. A. Shelankov, G. Pikus, “Reciprocity in reflection and transmission of light,” Phys. Rev. B 46, 3326–3336 (1992).
    [CrossRef]
  10. W. C. Snyder, “Reciprocity of the bidirectional reflectance distribution function (BRDF) in measurements and models of structured surfaces,” IEEE Trans. Geosci. Remote Sens. 36, 685–691 (1998).
    [CrossRef]
  11. C. von Fragstein, “Ist eine Lichtbewegung stets umkehrbar?” Opt. Acta 2, 16–22 (1955).
    [CrossRef]
  12. H. Okayama, I. Ogura, “Experimental verification of nonreciprocal response in light scattering from rough surfaces,” Appl. Opt. 23, 3349–3352 (1984).
    [CrossRef] [PubMed]
  13. W. H. Venable, “Comments on reciprocity failure,” Appl. Opt. 24, 3943 (1985).
    [CrossRef] [PubMed]
  14. M.-J. Kim, “Verification of the reciprocity theorem,” Appl. Opt. 27, 2645–2646 (1988).
    [CrossRef] [PubMed]
  15. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), pp. 380–381.
  16. A. T. DeHoop, “Reciprocity theorem for the electromagnetic field scattered by an obstacle,” Appl. Sci. Res. Sec. B 8, 135–140 (1960).
    [CrossRef]
  17. F. E. Nicodemus, “Directional reflectance and emissivity of an opaque surface,” Appl. Opt. 4, 767–773 (1965).
    [CrossRef]
  18. F. Nicodemus, Radiometry, Vol. 4 of Applied Optics and Optical Engineering (Academic, New York, 1967), Chap. 8, p. 284.
  19. G. Bauer, “Reflexionsmessungen an offenen Hohlräumen,” Optik 18, 603–622 (1961).
  20. K. Frank, Principles of Heat Transfer (Intext, New York, 1973), p. 244.
  21. A. Baltes, “On the validity of Kirchhoff’s law of heat radiation for a body in a nonequilibrium environment,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1976), Vol. 13, pp. 3–25.
    [CrossRef]
  22. J. Salisbury, A. Wald, D. D’Aria, “Thermal-infrared remote sensing and Kirchhoff’s law. 1. Laboratory measurements,” J. Geophys. Res. 99, 11897–11911 (1994).
    [CrossRef]
  23. F. Grum, R. Becherer, “Radiometry,” in Optical Radiation Measurements, F. Grum, ed. (Academic, New York, 1979), Vol. 1, p. 115.
  24. A. Springsteen, Labsphere, Inc., North Sutton, N.H. 03260-0070 (personal communication, 1996).
  25. D. S. Kimes, W. W. Newcomb, R. F. Nelson, J. B. Schutt, “Directional reflectance distributions of a hardwood and pine forest canopy,” IEEE Trans. Geosci. Remote Sens. GE-24, 281–293 (1986).
    [CrossRef]
  26. D. W. Deering, S. P. Ahmad, T. F. Eck, B. P. Banerjee, “Temporal attributes of the bidirectional reflectance for three Boreal forest canopies,” Proc. IGARSS’95 1, 1239–1241 (1995).
  27. K.-T. Kriebel, “On the limited validity of reciprocity in measured BRDFs,” Remote Sens. Environ. 58, 52–62 (1996).
    [CrossRef]

1998 (2)

W. C. Snyder, Z. Wan, “BRDF models to predict spectral reflectance and emissivity in the thermal infrared,” IEEE Trans. Geosci. Remote Sens. 36, 214–225 (1998).
[CrossRef]

W. C. Snyder, “Reciprocity of the bidirectional reflectance distribution function (BRDF) in measurements and models of structured surfaces,” IEEE Trans. Geosci. Remote Sens. 36, 685–691 (1998).
[CrossRef]

1996 (2)

1995 (1)

D. W. Deering, S. P. Ahmad, T. F. Eck, B. P. Banerjee, “Temporal attributes of the bidirectional reflectance for three Boreal forest canopies,” Proc. IGARSS’95 1, 1239–1241 (1995).

1994 (1)

J. Salisbury, A. Wald, D. D’Aria, “Thermal-infrared remote sensing and Kirchhoff’s law. 1. Laboratory measurements,” J. Geophys. Res. 99, 11897–11911 (1994).
[CrossRef]

1992 (1)

A. Shelankov, G. Pikus, “Reciprocity in reflection and transmission of light,” Phys. Rev. B 46, 3326–3336 (1992).
[CrossRef]

1988 (1)

1986 (1)

D. S. Kimes, W. W. Newcomb, R. F. Nelson, J. B. Schutt, “Directional reflectance distributions of a hardwood and pine forest canopy,” IEEE Trans. Geosci. Remote Sens. GE-24, 281–293 (1986).
[CrossRef]

1985 (2)

F. Clarke, D. Parry, “Helmholtz reciprocity: its validity and application to reflectometry,” Ltg. Res. Technol. 17, 1–11 (1985).
[CrossRef]

W. H. Venable, “Comments on reciprocity failure,” Appl. Opt. 24, 3943 (1985).
[CrossRef] [PubMed]

1984 (1)

1965 (1)

1961 (1)

G. Bauer, “Reflexionsmessungen an offenen Hohlräumen,” Optik 18, 603–622 (1961).

1960 (1)

A. T. DeHoop, “Reciprocity theorem for the electromagnetic field scattered by an obstacle,” Appl. Sci. Res. Sec. B 8, 135–140 (1960).
[CrossRef]

1955 (2)

C. von Fragstein, “Über die Formulierung des Kirchhoffschen Gesetzes und ihre Bedeutung für eine zweckmässige Definition von Remissionszahlen,” Optik 12, 60–68 (1955).

C. von Fragstein, “Ist eine Lichtbewegung stets umkehrbar?” Opt. Acta 2, 16–22 (1955).
[CrossRef]

Ahmad, S. P.

D. W. Deering, S. P. Ahmad, T. F. Eck, B. P. Banerjee, “Temporal attributes of the bidirectional reflectance for three Boreal forest canopies,” Proc. IGARSS’95 1, 1239–1241 (1995).

Baltes, A.

A. Baltes, “On the validity of Kirchhoff’s law of heat radiation for a body in a nonequilibrium environment,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1976), Vol. 13, pp. 3–25.
[CrossRef]

Banerjee, B. P.

D. W. Deering, S. P. Ahmad, T. F. Eck, B. P. Banerjee, “Temporal attributes of the bidirectional reflectance for three Boreal forest canopies,” Proc. IGARSS’95 1, 1239–1241 (1995).

Bauer, G.

G. Bauer, “Reflexionsmessungen an offenen Hohlräumen,” Optik 18, 603–622 (1961).

Becherer, R.

F. Grum, R. Becherer, “Radiometry,” in Optical Radiation Measurements, F. Grum, ed. (Academic, New York, 1979), Vol. 1, p. 115.

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), pp. 380–381.

Clarke, F.

F. Clarke, D. Parry, “Helmholtz reciprocity: its validity and application to reflectometry,” Ltg. Res. Technol. 17, 1–11 (1985).
[CrossRef]

D’Aria, D.

J. Salisbury, A. Wald, D. D’Aria, “Thermal-infrared remote sensing and Kirchhoff’s law. 1. Laboratory measurements,” J. Geophys. Res. 99, 11897–11911 (1994).
[CrossRef]

Deering, D. W.

D. W. Deering, S. P. Ahmad, T. F. Eck, B. P. Banerjee, “Temporal attributes of the bidirectional reflectance for three Boreal forest canopies,” Proc. IGARSS’95 1, 1239–1241 (1995).

DeHoop, A. T.

A. T. DeHoop, “Reciprocity theorem for the electromagnetic field scattered by an obstacle,” Appl. Sci. Res. Sec. B 8, 135–140 (1960).
[CrossRef]

Eck, T. F.

D. W. Deering, S. P. Ahmad, T. F. Eck, B. P. Banerjee, “Temporal attributes of the bidirectional reflectance for three Boreal forest canopies,” Proc. IGARSS’95 1, 1239–1241 (1995).

Frank, K.

K. Frank, Principles of Heat Transfer (Intext, New York, 1973), p. 244.

Ginsberg, I.

F. Nicodemus, J. Richmond, J. Hsia, I. Ginsberg, T. Limperis, Geometrical Considerations and Nomenclature for Reflectance, NBS Monograph 160 (National Bureau of Standards, Washington, D.C., 1977), pp. 3–6.

Grum, F.

F. Grum, R. Becherer, “Radiometry,” in Optical Radiation Measurements, F. Grum, ed. (Academic, New York, 1979), Vol. 1, p. 115.

Howell, J. R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer, 2nd ed. (Hemisphere, Washington, D.C., 1981), pp. 65–66.

Hsia, J.

F. Nicodemus, J. Richmond, J. Hsia, I. Ginsberg, T. Limperis, Geometrical Considerations and Nomenclature for Reflectance, NBS Monograph 160 (National Bureau of Standards, Washington, D.C., 1977), pp. 3–6.

Kim, M.-J.

Kimes, D. S.

D. S. Kimes, W. W. Newcomb, R. F. Nelson, J. B. Schutt, “Directional reflectance distributions of a hardwood and pine forest canopy,” IEEE Trans. Geosci. Remote Sens. GE-24, 281–293 (1986).
[CrossRef]

Kriebel, K.-T.

K.-T. Kriebel, “On the limited validity of reciprocity in measured BRDFs,” Remote Sens. Environ. 58, 52–62 (1996).
[CrossRef]

Levi, L.

L. Levi, Applied Optics (Wiley, New York, 1968), p. 84.

Limperis, T.

F. Nicodemus, J. Richmond, J. Hsia, I. Ginsberg, T. Limperis, Geometrical Considerations and Nomenclature for Reflectance, NBS Monograph 160 (National Bureau of Standards, Washington, D.C., 1977), pp. 3–6.

Nelson, R. F.

D. S. Kimes, W. W. Newcomb, R. F. Nelson, J. B. Schutt, “Directional reflectance distributions of a hardwood and pine forest canopy,” IEEE Trans. Geosci. Remote Sens. GE-24, 281–293 (1986).
[CrossRef]

Newcomb, W. W.

D. S. Kimes, W. W. Newcomb, R. F. Nelson, J. B. Schutt, “Directional reflectance distributions of a hardwood and pine forest canopy,” IEEE Trans. Geosci. Remote Sens. GE-24, 281–293 (1986).
[CrossRef]

Nicodemus, F.

F. Nicodemus, J. Richmond, J. Hsia, I. Ginsberg, T. Limperis, Geometrical Considerations and Nomenclature for Reflectance, NBS Monograph 160 (National Bureau of Standards, Washington, D.C., 1977), pp. 3–6.

F. Nicodemus, Radiometry, Vol. 4 of Applied Optics and Optical Engineering (Academic, New York, 1967), Chap. 8, p. 284.

Nicodemus, F. E.

Ogura, I.

Okayama, H.

Parry, D.

F. Clarke, D. Parry, “Helmholtz reciprocity: its validity and application to reflectometry,” Ltg. Res. Technol. 17, 1–11 (1985).
[CrossRef]

Pikus, G.

A. Shelankov, G. Pikus, “Reciprocity in reflection and transmission of light,” Phys. Rev. B 46, 3326–3336 (1992).
[CrossRef]

Reif, F.

F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965), pp. 382–384.

Richmond, J.

F. Nicodemus, J. Richmond, J. Hsia, I. Ginsberg, T. Limperis, Geometrical Considerations and Nomenclature for Reflectance, NBS Monograph 160 (National Bureau of Standards, Washington, D.C., 1977), pp. 3–6.

Salisbury, J.

J. Salisbury, A. Wald, D. D’Aria, “Thermal-infrared remote sensing and Kirchhoff’s law. 1. Laboratory measurements,” J. Geophys. Res. 99, 11897–11911 (1994).
[CrossRef]

Schutt, J. B.

D. S. Kimes, W. W. Newcomb, R. F. Nelson, J. B. Schutt, “Directional reflectance distributions of a hardwood and pine forest canopy,” IEEE Trans. Geosci. Remote Sens. GE-24, 281–293 (1986).
[CrossRef]

Shelankov, A.

A. Shelankov, G. Pikus, “Reciprocity in reflection and transmission of light,” Phys. Rev. B 46, 3326–3336 (1992).
[CrossRef]

Siegel, R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer, 2nd ed. (Hemisphere, Washington, D.C., 1981), pp. 65–66.

Snyder, W. C.

W. C. Snyder, “Reciprocity of the bidirectional reflectance distribution function (BRDF) in measurements and models of structured surfaces,” IEEE Trans. Geosci. Remote Sens. 36, 685–691 (1998).
[CrossRef]

W. C. Snyder, Z. Wan, “BRDF models to predict spectral reflectance and emissivity in the thermal infrared,” IEEE Trans. Geosci. Remote Sens. 36, 214–225 (1998).
[CrossRef]

W. C. Snyder, Z. Wan, “Surface temperature correction for active infrared reflectance measurements of natural materials,” Appl. Opt. 35, 2216–2220 (1996).
[CrossRef] [PubMed]

Springsteen, A.

A. Springsteen, Labsphere, Inc., North Sutton, N.H. 03260-0070 (personal communication, 1996).

Venable, W. H.

von Fragstein, C.

C. von Fragstein, “Ist eine Lichtbewegung stets umkehrbar?” Opt. Acta 2, 16–22 (1955).
[CrossRef]

C. von Fragstein, “Über die Formulierung des Kirchhoffschen Gesetzes und ihre Bedeutung für eine zweckmässige Definition von Remissionszahlen,” Optik 12, 60–68 (1955).

Wald, A.

J. Salisbury, A. Wald, D. D’Aria, “Thermal-infrared remote sensing and Kirchhoff’s law. 1. Laboratory measurements,” J. Geophys. Res. 99, 11897–11911 (1994).
[CrossRef]

Wan, Z.

W. C. Snyder, Z. Wan, “BRDF models to predict spectral reflectance and emissivity in the thermal infrared,” IEEE Trans. Geosci. Remote Sens. 36, 214–225 (1998).
[CrossRef]

W. C. Snyder, Z. Wan, “Surface temperature correction for active infrared reflectance measurements of natural materials,” Appl. Opt. 35, 2216–2220 (1996).
[CrossRef] [PubMed]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), pp. 380–381.

Appl. Opt. (5)

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

A. T. DeHoop, “Reciprocity theorem for the electromagnetic field scattered by an obstacle,” Appl. Sci. Res. Sec. B 8, 135–140 (1960).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (3)

W. C. Snyder, Z. Wan, “BRDF models to predict spectral reflectance and emissivity in the thermal infrared,” IEEE Trans. Geosci. Remote Sens. 36, 214–225 (1998).
[CrossRef]

W. C. Snyder, “Reciprocity of the bidirectional reflectance distribution function (BRDF) in measurements and models of structured surfaces,” IEEE Trans. Geosci. Remote Sens. 36, 685–691 (1998).
[CrossRef]

D. S. Kimes, W. W. Newcomb, R. F. Nelson, J. B. Schutt, “Directional reflectance distributions of a hardwood and pine forest canopy,” IEEE Trans. Geosci. Remote Sens. GE-24, 281–293 (1986).
[CrossRef]

J. Geophys. Res. (1)

J. Salisbury, A. Wald, D. D’Aria, “Thermal-infrared remote sensing and Kirchhoff’s law. 1. Laboratory measurements,” J. Geophys. Res. 99, 11897–11911 (1994).
[CrossRef]

Ltg. Res. Technol. (1)

F. Clarke, D. Parry, “Helmholtz reciprocity: its validity and application to reflectometry,” Ltg. Res. Technol. 17, 1–11 (1985).
[CrossRef]

Opt. Acta (1)

C. von Fragstein, “Ist eine Lichtbewegung stets umkehrbar?” Opt. Acta 2, 16–22 (1955).
[CrossRef]

Optik (2)

C. von Fragstein, “Über die Formulierung des Kirchhoffschen Gesetzes und ihre Bedeutung für eine zweckmässige Definition von Remissionszahlen,” Optik 12, 60–68 (1955).

G. Bauer, “Reflexionsmessungen an offenen Hohlräumen,” Optik 18, 603–622 (1961).

Phys. Rev. B (1)

A. Shelankov, G. Pikus, “Reciprocity in reflection and transmission of light,” Phys. Rev. B 46, 3326–3336 (1992).
[CrossRef]

Proc. IGARSS’95 (1)

D. W. Deering, S. P. Ahmad, T. F. Eck, B. P. Banerjee, “Temporal attributes of the bidirectional reflectance for three Boreal forest canopies,” Proc. IGARSS’95 1, 1239–1241 (1995).

Remote Sens. Environ. (1)

K.-T. Kriebel, “On the limited validity of reciprocity in measured BRDFs,” Remote Sens. Environ. 58, 52–62 (1996).
[CrossRef]

Other (10)

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer, 2nd ed. (Hemisphere, Washington, D.C., 1981), pp. 65–66.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), pp. 380–381.

K. Frank, Principles of Heat Transfer (Intext, New York, 1973), p. 244.

A. Baltes, “On the validity of Kirchhoff’s law of heat radiation for a body in a nonequilibrium environment,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1976), Vol. 13, pp. 3–25.
[CrossRef]

F. Grum, R. Becherer, “Radiometry,” in Optical Radiation Measurements, F. Grum, ed. (Academic, New York, 1979), Vol. 1, p. 115.

A. Springsteen, Labsphere, Inc., North Sutton, N.H. 03260-0070 (personal communication, 1996).

F. Nicodemus, Radiometry, Vol. 4 of Applied Optics and Optical Engineering (Academic, New York, 1967), Chap. 8, p. 284.

L. Levi, Applied Optics (Wiley, New York, 1968), p. 84.

F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965), pp. 382–384.

F. Nicodemus, J. Richmond, J. Hsia, I. Ginsberg, T. Limperis, Geometrical Considerations and Nomenclature for Reflectance, NBS Monograph 160 (National Bureau of Standards, Washington, D.C., 1977), pp. 3–6.

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Figures (3)

Fig. 1
Fig. 1

Geometry definitions for the BRDF on the unit hemisphere.

Fig. 2
Fig. 2

Form of the Faraday isolator. The S polarization is rotated to line up with the S′ polarization for right-to-left propagation. For left-to-right propagation, the S′ is rotated to the P polarization and is blocked.

Fig. 3
Fig. 3

Structured system that violates reciprocity and the directional form of Kirchhoff’s law. Straight arrows show radiation from the cavity walls that has not been absorbed by the isolator. The wavy arrows show the radiation emitted by the isolator. The S polarization from the b branch reflects without being absorbed with an S′ polarization toward the a branch. The P polarization from the b branch and both the S′ and P′ polarizations from the a branch are absorbed by the isolator.

Equations (24)

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f θ i ,   ϕ i ;   θ j ,   ϕ j = f θ j ,   ϕ j ;   θ i ,   ϕ i .
d ω d A i = 1 R 2 = 1 .
d 2 Φ = Ld ω dA   cos   θ .
f θ i ,   ϕ i ;   θ j ,   ϕ j = dL j dE i .
d 3 Φ = f θ i ,   ϕ i ;   θ j ,   ϕ j dA f   cos   θ j dA j L b dA i   cos   θ i .
d 2 Φ = 2 π   f θ i ,   ϕ i ;   θ j ,   ϕ j dA f   cos   θ j dA j L b d Ω i ,
d 2 Φ = ε θ j ,   ϕ j L b dA f dA j   cos   θ j .
d 2 Φ = 2 π   f θ i ,   ϕ i ;   θ j ,   ϕ j dA f   cos   θ j dA j L b d Ω i + ε θ j ,   ϕ j L b dA f dA j   cos   θ j .
d 2 Φ = L b dA f dA j   cos   θ j .
Ω a d Ω j = Ω a 2 π   f i ,   j d Ω i d Ω j + Ω a   ε j d Ω j .
Ω b d Ω j = Ω b 2 π   f i ,   j d Ω i d Ω j + Ω b   ε j d Ω j .
Ω a d Ω j     Ω b d Ω j     k ,
Ω a 2 π   f i ,   j d Ω i d Ω j + Ω a   ε j d Ω j = Ω b 2 π   f i ,   j d Ω i d Ω j + Ω b   ε j d Ω j .
Ω a 2 π   f i ,   j d Ω i d Ω j = Ω a Ω b   f i ,   j d Ω i + Ω b \ 2 π   f i ,   j d Ω i d Ω j .
Ω a Ω b \ 2 π   f i ,   j d Ω i d Ω j     Ω b Ω a \ 2 π   f i ,   j d Ω i d Ω j .
Ω a Ω b   f i ,   j d Ω i d Ω j + Ω a   ε j d Ω j =   Ω b Ω a   f i ,   j d Ω i d Ω j + Ω b   ε j d Ω j .
f b ,   a Ω a d Ω j Ω b d Ω i + ε a Ω a d Ω j =   f a ,   b Ω b d Ω j Ω a d Ω i + ε b Ω b d Ω j .
ε θ a ,   ϕ a - ε θ b ,   ϕ b = k f θ a ,   ϕ a ;   θ b ,   ϕ b - f θ b ,   ϕ b ;   θ a ,   ϕ a .
ε θ j ,   ϕ j = 1 - 2 π   f θ i ,   ϕ i ;   θ j ,   ϕ j d Ω i .
α θ j ,   ϕ j = 1 - 2 π   f θ j ,   ϕ j ;   θ i ,   ϕ i d Ω i .
ε θ j ,   ϕ j = α θ j ,   ϕ j ,
2 π   f θ i ,   ϕ i ;   θ j ,   ϕ j d Ω i = 2 π   f θ j ,   ϕ j ;   θ i ,   ϕ i d Ω i .
f λ ,   φ ;   θ i ,   ϕ i ;   θ j ,   ϕ j = f λ ,   φ ;   θ j ,   ϕ j ;   θ i ,   ϕ i ,
ε λ ,   φ ;   θ j ,   ϕ j = α λ ,   φ ;   θ j ,   ϕ j .

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