Abstract

I address the question of whether the bidirectional reflectance distribution function (BRDF) of a structured surface has the same value when the incident and reflected angles are reversed. In particular, I examine the validity of some recent counterexamples and contrary measurements. On the basis of a new definition for structured surfaces, I conclude that the BRDF should be reciprocal. I show that designed counterexamples are flawed and suggest that measurements that do not exhibit reciprocity can be attributed to uncertainty and uncontrolled factors.

© 2002 Optical Society of America

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References

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  1. F. E. Nicodemus, Radiometry, Vol. 4 of the Applied Optics and Optical Engineering Series (Academic, New York, 1967).
  2. F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Monogr. 160, (1977).
  3. W. G. Rees, Physical Principles of Remote Sensing (Cambridge U. Press, Cambridge, UK, 1990).
  4. F. J. J. Clarke, D. J. Parry, “Helmholtz reciprocity: its validity and application to reflectometry,” Light Res. Technol. 17, 1–11 (1985).
    [CrossRef]
  5. R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer, 2nd ed. (Hemisphere, Washington, D.C., 1981).
  6. W. C. Snyder, Z. Wan, X. Li, “Thermodynamic constraints on reflectance reciprocity and Kirchhoff’s law,” Appl. Opt. 37, 3464–3470 (1998).
    [CrossRef]
  7. R. Carminati, M. Nieto-Vesperinas, J.-J. Greffet, “Reciprocity of evanescent electromagnetic waves,” J. Opt. Soc. Am. A 15, 706–712 (1998).
    [CrossRef]
  8. J.-J. Greffet, M. Nieto-Vesperinas, “Field theory for generalized bidirectional reflectivity: derivation of Helmholtz’s reciprocity principle and Kirchhoff’s law,” J. Opt. Soc. Am. A 15, 2735–2744 (1998).
    [CrossRef]
  9. 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]
  10. W. C. Snyder, “Definition and invariance properties of structured surface BRDF,” IEEE Trans. Geosci. Remote Sens. (to be published).
  11. H. Okayama, I. Ogura, “Experimental verification of nonreciprocal response in light scattering from rough surfaces,” Appl. Opt. 23, 3349–3352 (1984).
    [CrossRef] [PubMed]
  12. W. Venable, “Comments on reciprocity failure,” Appl. Opt. 24, 3943 (1985).
    [CrossRef] [PubMed]
  13. H. J. Eom, “Energy conservation and reciprocity of random rough surface scattering,” Appl. Opt. 24, 1730–1732 (1985).
    [CrossRef] [PubMed]
  14. M.-J. Kim, “Verification of the reciprocity theorem,” Appl. Opt. 27, 2645–2646 (1988).
    [CrossRef] [PubMed]
  15. X. Li, Z. Wan, “Comments on reciprocity in the directional reflectance modeling,” Prog. Nat. Sci. 8, 354–358 (1998).
  16. E. Eisner, “Acoustic reciprocity paradox,” Geophysics 48, 1132–1134 (1981).
    [CrossRef]
  17. X. Li, J. Wang, A. Strahler, “Apparent reciprocity failure in directional reflectance of structured surfaces,” Prog. Nat. Sci. 9, 747–752 (1999).
  18. F. A. Dahlen, R. I. Odom, “Acoustic reciprocity—a paradox,” Geophysics 49, 478–480 (1984).
    [CrossRef]
  19. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
    [CrossRef]
  20. M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light (Cambridge U. Press, Cambridge, UK, 1990).

1999 (1)

X. Li, J. Wang, A. Strahler, “Apparent reciprocity failure in directional reflectance of structured surfaces,” Prog. Nat. Sci. 9, 747–752 (1999).

1998 (5)

1988 (1)

1985 (3)

1984 (2)

1981 (1)

E. Eisner, “Acoustic reciprocity paradox,” Geophysics 48, 1132–1134 (1981).
[CrossRef]

1977 (1)

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Monogr. 160, (1977).

Born, M.

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light (Cambridge U. Press, Cambridge, UK, 1990).

Carminati, R.

Clarke, F. J. J.

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

Dahlen, F. A.

F. A. Dahlen, R. I. Odom, “Acoustic reciprocity—a paradox,” Geophysics 49, 478–480 (1984).
[CrossRef]

Eisner, E.

E. Eisner, “Acoustic reciprocity paradox,” Geophysics 48, 1132–1134 (1981).
[CrossRef]

Eom, H. J.

Ginsberg, I. W.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Monogr. 160, (1977).

Greffet, J.-J.

Howell, J. R.

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

Hsia, J. J.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Monogr. 160, (1977).

Kim, M.-J.

Li, X.

X. Li, J. Wang, A. Strahler, “Apparent reciprocity failure in directional reflectance of structured surfaces,” Prog. Nat. Sci. 9, 747–752 (1999).

W. C. Snyder, Z. Wan, X. Li, “Thermodynamic constraints on reflectance reciprocity and Kirchhoff’s law,” Appl. Opt. 37, 3464–3470 (1998).
[CrossRef]

X. Li, Z. Wan, “Comments on reciprocity in the directional reflectance modeling,” Prog. Nat. Sci. 8, 354–358 (1998).

Limperis, T.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Monogr. 160, (1977).

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
[CrossRef]

Nicodemus, F. E.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Monogr. 160, (1977).

F. E. Nicodemus, Radiometry, Vol. 4 of the Applied Optics and Optical Engineering Series (Academic, New York, 1967).

Nieto-Vesperinas, M.

Odom, R. I.

F. A. Dahlen, R. I. Odom, “Acoustic reciprocity—a paradox,” Geophysics 49, 478–480 (1984).
[CrossRef]

Ogura, I.

Okayama, H.

Parry, D. J.

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

Rees, W. G.

W. G. Rees, Physical Principles of Remote Sensing (Cambridge U. Press, Cambridge, UK, 1990).

Richmond, J. C.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Monogr. 160, (1977).

Siegel, R.

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

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, X. Li, “Thermodynamic constraints on reflectance reciprocity and Kirchhoff’s law,” Appl. Opt. 37, 3464–3470 (1998).
[CrossRef]

W. C. Snyder, “Definition and invariance properties of structured surface BRDF,” IEEE Trans. Geosci. Remote Sens. (to be published).

Strahler, A.

X. Li, J. Wang, A. Strahler, “Apparent reciprocity failure in directional reflectance of structured surfaces,” Prog. Nat. Sci. 9, 747–752 (1999).

Venable, W.

Wan, Z.

X. Li, Z. Wan, “Comments on reciprocity in the directional reflectance modeling,” Prog. Nat. Sci. 8, 354–358 (1998).

W. C. Snyder, Z. Wan, X. Li, “Thermodynamic constraints on reflectance reciprocity and Kirchhoff’s law,” Appl. Opt. 37, 3464–3470 (1998).
[CrossRef]

Wang, J.

X. Li, J. Wang, A. Strahler, “Apparent reciprocity failure in directional reflectance of structured surfaces,” Prog. Nat. Sci. 9, 747–752 (1999).

Wolf, E.

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light (Cambridge U. Press, Cambridge, UK, 1990).

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
[CrossRef]

Appl. Opt. (5)

Geophysics (2)

E. Eisner, “Acoustic reciprocity paradox,” Geophysics 48, 1132–1134 (1981).
[CrossRef]

F. A. Dahlen, R. I. Odom, “Acoustic reciprocity—a paradox,” Geophysics 49, 478–480 (1984).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

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]

J. Opt. Soc. Am. A (2)

Light Res. Technol. (1)

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

Natl. Bur. Stand. (U.S.) Monogr. (1)

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Monogr. 160, (1977).

Prog. Nat. Sci. (2)

X. Li, Z. Wan, “Comments on reciprocity in the directional reflectance modeling,” Prog. Nat. Sci. 8, 354–358 (1998).

X. Li, J. Wang, A. Strahler, “Apparent reciprocity failure in directional reflectance of structured surfaces,” Prog. Nat. Sci. 9, 747–752 (1999).

Other (6)

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
[CrossRef]

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light (Cambridge U. Press, Cambridge, UK, 1990).

F. E. Nicodemus, Radiometry, Vol. 4 of the Applied Optics and Optical Engineering Series (Academic, New York, 1967).

W. C. Snyder, “Definition and invariance properties of structured surface BRDF,” IEEE Trans. Geosci. Remote Sens. (to be published).

W. G. Rees, Physical Principles of Remote Sensing (Cambridge U. Press, Cambridge, UK, 1990).

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

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Figures (5)

Fig. 1
Fig. 1

Cross section of Eisner’s paradox—a truncated ellipsoid with a source and detector at the foci. The truncation boundary subtends a much smaller solid angle that can be seen from position 1, compared with position 2.

Fig. 2
Fig. 2

Ellipsoidal section surface with a source and detector at the foci of a section. This is not reciprocal for a finite-sized source and detector.

Fig. 3
Fig. 3

Cross section of one cell of an array of telescopes. The primary mirror at the lower right reflects all parallel rays from direction 1 to the secondary and off in direction 2. On the other hand, most of the parallel rays from 2 would miss the secondary and not travel to direction 1 by the reverse paths. This reasoning, however, cannot be used to show a reciprocity violation (see text).

Fig. 4
Fig. 4

Definitions for the axial, zero-power telescope configuration.

Fig. 5
Fig. 5

Geometric arrangement and definitions for the relation between flux and BRDF reciprocity.

Equations (23)

Equations on this page are rendered with MathJax. Learn more.

θ1=arccos21+2,
Ω=2π 1+cos θ1.
Φ2Φ1= 1+cos θ1=1+21+2+21+2=1+21+2.
E= L cos θsin θdθdϕ.
L2=qE CP/CSCS/C=qE CP/C.
L1=qE CP/C.
L1E=L2E,
f12=f21.
M=F1F2.
R=R1M2+R2.
Φd=LsAsAdR2.
As=AsM2.
Φ12=M2LsAsAdR1+R2M22.
Φ21=LsM2AsAdM2R2+R12,
Φ12=Φ21.
f12=dL2dE1L2E1.
L12=Φ12R22A cos θ2Ad,
L21=Φ21R12A cos θ1Ad.
E12=LsAs cos θ1R12,
E21=LsAs cos θ2R22.
f12= L12E12=Φ12R22R12LsAs cos θ1A cos θ2Ad,
f21= L21E21=Φ21R12R22LsAs cos θ2A cos θ1Ad.
f12=f21.

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