Abstract

The impetus for this paper was to assist investigators not familiar with radiometry theory. Many times they apply a scalar theory to the scattering of polarized light from nondepolarizing targets. Also some investigators erroneously apply a combined scalar–vector theory since the correct vector–matrix approach has not been developed in the literature. The classical theory used by many investigators is modified in vector–matrix form in this paper. The vector–matrix theory is applied to the various geometries. These results and examples are presented in tabular form.

© 1991 Optical Society of America

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References

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  1. F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical Considerations and Nomenclature for Reflectance,” Self Study Manual on Optical Radiation Measurements, NBS Monograph 160 (1977), Part 1, Chap. 6.
  2. W. L. Wolfe, Radiation Theory, The Infrared Handbook, ONR, Dept. of Navy, Chapter 1, 1–41 (1978).
  3. J. E. Harvey, Light Scattering Characteristics of Optical Surfaces, Ph.D. Thesis, University of Arizona (1979).
  4. W. S. Bickel, V. Iafelice, Polarized light scattering from surfaces, Proc. of the 1985 Sci. Conf. on Obscuration Aerosol Res., Report CRDEC-SP-86019 (1986).
  5. W. A. Shurcliff, Polarized Light (Harvard U. P., Cambridge, 1966).
  6. W. S. Bickel, W. M. Bailey, “Stokes Vectors, Mueller Matrices, and Polarized Scattered Light,” Am. J. Phys. 53, 468–478 (1985).
    [CrossRef]
  7. W. S. Bickel, R. R. Zato, V. J. Iafelice, “Polarized Light Scattered from Fundamental Surfaces,” J. Appl. Phys. 61, 5392–5398 (1987).
    [CrossRef]

1987 (1)

W. S. Bickel, R. R. Zato, V. J. Iafelice, “Polarized Light Scattered from Fundamental Surfaces,” J. Appl. Phys. 61, 5392–5398 (1987).
[CrossRef]

1985 (1)

W. S. Bickel, W. M. Bailey, “Stokes Vectors, Mueller Matrices, and Polarized Scattered Light,” Am. J. Phys. 53, 468–478 (1985).
[CrossRef]

Bailey, W. M.

W. S. Bickel, W. M. Bailey, “Stokes Vectors, Mueller Matrices, and Polarized Scattered Light,” Am. J. Phys. 53, 468–478 (1985).
[CrossRef]

Bickel, W. S.

W. S. Bickel, R. R. Zato, V. J. Iafelice, “Polarized Light Scattered from Fundamental Surfaces,” J. Appl. Phys. 61, 5392–5398 (1987).
[CrossRef]

W. S. Bickel, W. M. Bailey, “Stokes Vectors, Mueller Matrices, and Polarized Scattered Light,” Am. J. Phys. 53, 468–478 (1985).
[CrossRef]

W. S. Bickel, V. Iafelice, Polarized light scattering from surfaces, Proc. of the 1985 Sci. Conf. on Obscuration Aerosol Res., Report CRDEC-SP-86019 (1986).

Ginsberg, I. W.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical Considerations and Nomenclature for Reflectance,” Self Study Manual on Optical Radiation Measurements, NBS Monograph 160 (1977), Part 1, Chap. 6.

Harvey, J. E.

J. E. Harvey, Light Scattering Characteristics of Optical Surfaces, Ph.D. Thesis, University of Arizona (1979).

Hsia, J. J.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical Considerations and Nomenclature for Reflectance,” Self Study Manual on Optical Radiation Measurements, NBS Monograph 160 (1977), Part 1, Chap. 6.

Iafelice, V.

W. S. Bickel, V. Iafelice, Polarized light scattering from surfaces, Proc. of the 1985 Sci. Conf. on Obscuration Aerosol Res., Report CRDEC-SP-86019 (1986).

Iafelice, V. J.

W. S. Bickel, R. R. Zato, V. J. Iafelice, “Polarized Light Scattered from Fundamental Surfaces,” J. Appl. Phys. 61, 5392–5398 (1987).
[CrossRef]

Limperis, T.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical Considerations and Nomenclature for Reflectance,” Self Study Manual on Optical Radiation Measurements, NBS Monograph 160 (1977), Part 1, Chap. 6.

Nicodemus, F. E.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical Considerations and Nomenclature for Reflectance,” Self Study Manual on Optical Radiation Measurements, NBS Monograph 160 (1977), Part 1, Chap. 6.

Richmond, J. C.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical Considerations and Nomenclature for Reflectance,” Self Study Manual on Optical Radiation Measurements, NBS Monograph 160 (1977), Part 1, Chap. 6.

Shurcliff, W. A.

W. A. Shurcliff, Polarized Light (Harvard U. P., Cambridge, 1966).

Wolfe, W. L.

W. L. Wolfe, Radiation Theory, The Infrared Handbook, ONR, Dept. of Navy, Chapter 1, 1–41 (1978).

Zato, R. R.

W. S. Bickel, R. R. Zato, V. J. Iafelice, “Polarized Light Scattered from Fundamental Surfaces,” J. Appl. Phys. 61, 5392–5398 (1987).
[CrossRef]

Am. J. Phys. (1)

W. S. Bickel, W. M. Bailey, “Stokes Vectors, Mueller Matrices, and Polarized Scattered Light,” Am. J. Phys. 53, 468–478 (1985).
[CrossRef]

J. Appl. Phys. (1)

W. S. Bickel, R. R. Zato, V. J. Iafelice, “Polarized Light Scattered from Fundamental Surfaces,” J. Appl. Phys. 61, 5392–5398 (1987).
[CrossRef]

Other (5)

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical Considerations and Nomenclature for Reflectance,” Self Study Manual on Optical Radiation Measurements, NBS Monograph 160 (1977), Part 1, Chap. 6.

W. L. Wolfe, Radiation Theory, The Infrared Handbook, ONR, Dept. of Navy, Chapter 1, 1–41 (1978).

J. E. Harvey, Light Scattering Characteristics of Optical Surfaces, Ph.D. Thesis, University of Arizona (1979).

W. S. Bickel, V. Iafelice, Polarized light scattering from surfaces, Proc. of the 1985 Sci. Conf. on Obscuration Aerosol Res., Report CRDEC-SP-86019 (1986).

W. A. Shurcliff, Polarized Light (Harvard U. P., Cambridge, 1966).

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Figures (4)

Fig. 1
Fig. 1

Geometry for measurement of BSSRDF.

Fig. 2
Fig. 2

Geometry for measurement of the bidirectional reflectance.

Fig. 3
Fig. 3

Sketch of irradiation geometries.

Fig. 4
Fig. 4

Schematic of a bistatic reflectometer.

Tables (4)

Tables Icon

Table I Emergent Flux for Various Geometries and Reflectance Matrices

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Table II Vector–Matrix Definition of Reflectance for the Nine Geometries of Nlcodemus et al.2 Incident

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Table III Reflectance for Nine Geometries for Constant and Nonconstant Incident Polarized Light for ArAi or dardai

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Table IV Values of the Bidirectional Reflectance Calculated by Taking the Ratio dϕr(θr,ϕr)/dϕi(θi,ϕi) of the First Components of the Detected Stokes Vector for Different Incident and Emergent Polarized States and for Specific Angles of Incidence and Emergence

Equations (39)

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L ¯ r ( θ i , ϕ i ; x i , y i ; θ r , ϕ r ; x r , y r ) = S ˆ ( θ i , ϕ i ; x i , y i ; θ r , ϕ r ; x r , y r ) d Φ ¯ i ( θ i , ϕ i ) ,
L ¯ r ( θ i , ϕ i ; x i , y i ; θ r , ϕ r ; x r , y r ) = [ A i S ˆ ( θ i , ϕ i ; x i , y i ; θ r , ϕ r ; x r , y r ) d A i ] × L ¯ i ( θ i , ϕ i ) cos θ i d ω i ,
f ˆ ( θ i , ϕ i ; θ r , ϕ r ; x r , y r ) = A i S ˆ ( θ i , ϕ i ; x i , y i ; θ r , ϕ r ; x r , y r ) d A i .
L ¯ ( θ i , ϕ i ; θ r , ϕ r ) = f ˆ ( θ i , ϕ i ; θ r , ϕ r ) L ¯ i ( θ i , ϕ i ) d Ω i .
d Φ ¯ r ( θ r , ϕ r ) = L ¯ r ( θ i , ϕ i ; θ r , ϕ r ) d Ω r d A r , d Φ ¯ r ( θ r , ϕ r ) = { [ f ˆ ( θ i , ϕ i ; θ r , ϕ r ) L ¯ r ( θ i , ϕ i ) d Ω i ] d Ω r } d A r .
Φ ¯ r ( ω r ) = ω r L ¯ r ( θ i , ϕ i ; θ r , ϕ r ) d Ω r A r = { ω r [ ω i f ˆ ( θ i , ϕ i ; θ r , ϕ r ) L ¯ i ( θ i , ϕ i ) d Ω i ] d Ω r } A r .
ρ ( θ i , ϕ i ; θ r , ϕ r ) = d Φ r ( θ r , ϕ r ) / d Φ i ( θ i , ϕ i ) .
d Φ ¯ r ( θ r , ϕ r ) = ρ ˆ ( θ i , ϕ i ; θ r , ϕ r ) d Φ ¯ i ( θ i , ϕ i ) .
d Φ ¯ r ( θ r , ϕ r ) = ρ ˆ ( θ i , ϕ i ; θ r , ϕ r ) L ¯ i ( θ i , ϕ i ) d Ω i d A i ,
d Φ ¯ r ( θ r , ϕ r ) = { [ f ˆ ( θ i , ϕ i ; θ r , ϕ r ) L ¯ i ( θ i , ϕ i ) d Ω i ] d Ω r } d A r ,
ρ ˆ ( θ i , ϕ i ; θ r , ϕ r ) L ¯ i = f ˆ ( θ i , ϕ i ; θ r , ϕ r ) L ¯ i d Ω r d A r / d A i ,
ρ ˆ ( θ i , ϕ i ; θ r , ϕ r ) L ¯ i = f ˆ ( θ i , ϕ i ; θ r , ϕ r ) L ¯ i d Ω r .
d Φ ¯ r ( θ r , ϕ r ) = d ρ ˆ ( θ i , ϕ i ; θ r , ϕ r ) L ¯ i ( θ i , ϕ i ) d A i
d Φ ¯ r ( θ r , ϕ r ) = ρ ˆ ( θ i , ϕ i ; θ r , ϕ r ) L ¯ i ( θ i , ϕ i ) d Ω i d A i .
Φ ¯ r ( θ r , ϕ r ) = { ω i [ d ρ ˆ ( θ i , ϕ i ; θ r , ϕ r ) L ¯ i ( θ i , ϕ i ) ] } A i ,
Φ ¯ r ( θ r , ϕ r ) = ω i [ ρ ˆ ( θ i , ϕ i ; θ r , ϕ r ) L ¯ i ( θ i , ϕ i ) d Ω i ] A i .
Φ ¯ r ( ω r ) = { ω r ω i [ d ρ ˆ ( θ i , ϕ i ; θ r , ϕ r ) L ¯ i ( θ i , ϕ i ) ] } A i ,
Φ ¯ r ( ω r ) = ω i ω r [ ρ ¯ ( θ i , ϕ i ; θ r , ϕ r ) L ¯ i ( θ i , ϕ i ) d Ω i d Ω r ] A i .
ρ ˆ ( θ i , ϕ i ; θ r , ϕ r ) d Ω i L ¯ i = f ˆ ( θ i , ϕ i ; θ r , ϕ r ) d Ω i d Ω r L ¯ i
ρ k l ( θ i , ϕ i ; θ r , ϕ r ) = f k l ( θ i , ϕ i ; θ r , ϕ r ) d Ω r .
ρ ˆ ( ω i ; ω r ) ( ω i d Ω i ) L ¯ i = { ω r [ ω i ρ ˆ ( θ i , ϕ i ; θ r , ϕ r ) d Ω i ] d Ω r } L ¯ i
ρ ˆ ( ω i ; ω r ) [ ω i d Ω i ] L ¯ i = { ω r [ ω i f ˆ ( θ i , ϕ i ; θ r , ϕ r ) d Ω i ] d Ω r } L ¯ i ;
ρ k l ( ω i ; ω r ) = 1 / ω i { ω r [ ω i ρ k l ( θ i , ϕ i ; θ r , ϕ r ) d Ω i ] d Ω r }
ρ k l ( ω i ; ω r ) = 1 / ω i { ω r [ ω i f k l ( θ i , ϕ i ; θ r , ϕ r ) d Ω i ] d Ω r } .
ρ ˆ ( 2 π ; 2 π ) ( 2 π d Ω i ) L ¯ i = { 2 π [ 2 π ρ ˆ ( θ i , ϕ i ; θ r , ϕ r ) d Ω i ] d Ω r } L ¯ i
ρ ˆ ( 2 π ; 2 π ) ( 2 π d Ω i ) L ¯ i = { 2 π [ 2 π f ˆ ( θ i , ϕ i ; θ r , ϕ r ) d Ω i ] d Ω r } L ¯ i ;
ρ k l ( 2 π ; 2 π ) = 1 / π { 2 π [ 2 π ρ k l ( θ i , ϕ i ; ϕ r , ϕ r ) d Ω i ] d Ω r }
ρ k l ( 2 π ; 2 π ) = 1 / π { 2 π [ 2 π f k l ( θ i , ϕ i ; ϕ r , ϕ r ) d Ω i ] d Ω r } .
[ ρ ˆ ( θ i , ϕ i ; θ r , ϕ r ) L ¯ i ( θ i , ϕ i ) ] s = [ f ˆ ( θ i , ϕ i ; θ r , ϕ r ) L ¯ i ( θ i , ϕ i ) d Ω r ] s .
[ ρ ˆ ( ω i ; ω r ) ω i L ¯ i ( θ ι , ϕ i ) d Ω i ] s = { ω r [ ω i ρ ˆ ( θ i , ϕ i ; ϕ r , ϕ r ) L ¯ i ( θ i , ϕ i ) d Ω i ] d Ω r } s
[ ρ ˆ ( ω i ; ω r ) ω i L ¯ i ( θ ι , ϕ i ) d Ω i ] s = { ω r [ ω i f ˆ ( θ i , ϕ i ; ϕ r , ϕ r ) L ¯ i ( θ i , ϕ i ) d Ω i ] d Ω r } s .
[ ρ ˆ ( 2 π ; 2 π ) 2 π L ¯ i ( θ ι , ϕ i ) d Ω i ] s = { 2 π [ 2 π ρ ˆ ( θ i , ϕ i ; ϕ r , ϕ r ) L ¯ i ( θ i , ϕ i ) d Ω i ] d Ω r } s
[ ρ ˆ ( 2 π ; 2 π ) 2 π L ¯ i ( θ ι , ϕ i ) d Ω i ] s = { 2 π [ 2 π f ˆ ( θ i , ϕ i ; ϕ r , ϕ r ) L ¯ i ( θ i , ϕ i ) d Ω i ] d Ω r } s .
d Φ ¯ d r ( θ r , ϕ r ) = A ˆ ρ ˆ ( θ i , ϕ i ; θ r , ϕ r ) P ˆ d Φ ¯ i ( θ i , ϕ i ) .
ρ ˆ = ( ρ 11 ρ l 2 ρ 13 ρ 14 ρ 21 ρ 22 ρ 23 ρ 24 ρ 31 p 32 ρ 33 ρ 34 ρ 41 ρ 42 p 43 ρ 44 ) ,
ρ = d Φ r ( θ r , ϕ r ) / d Φ i ( θ i , ϕ i ) .
d Φ ¯ d r ( θ r , ϕ r ) = A ¯ ρ ˆ ( θ i , ϕ i ; θ r , ϕ r ) P ˆ d Φ ¯ i ( θ i , ϕ i ) = 1 2 ( 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 ) ( ρ 11 ρ 12 ρ 13 ρ 14 ρ 21 ρ 22 ρ 23 ρ 24 ρ 31 ρ 32 ρ 33 ρ 34 ρ 41 ρ 42 ρ 43 ρ 44 ) ( d Φ i ( θ i , ϕ i ) 0 0 0 ) .
d Φ r ( θ r , ϕ r ) = 1 2 [ ρ 11 ( θ i , ϕ i ; θ r , ϕ r ) + ρ 21 ( θ i , ϕ i ; θ r , ϕ r ) ] d Φ i ( θ i , ϕ i ) .
d Φ r ( θ r , ϕ r ) = 1 2 [ ρ 11 ( θ i , ϕ i ; θ r , ϕ r ) + ρ 12 ( θ i , ϕ i ; θ r , ϕ r ) ] d Φ i ( θ i , ϕ i ) .

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