Abstract

Variable coherence tomography (VCT) was recently developed by Baleine and Dogariu for the purpose of directly sensing the second-order statistical properties of a randomly scattering volume [J. Opt. Soc. Am. A 21, 1917 (2004) ]. In this paper we generalize the theory of VCT to include polarized inputs and anisotropic scatterers. In general the measurement of the scattered coherency matrix or Stokes vector is not adequate to describe the scattering, as these quantities depend on the coherence state of the incident beam. However, by controlling the polarized coherence properties of the source beam, VCT can be generalized to probe the polarimetric scattering properties of objects from a single-point Stokes vector or coherency matrix measurements. With polarized VCT, we are able to design a method that can measure analogous information to the polarimetric bidirectional reflection distribution function (BRDF), but do it from monostatic data. This capability would allow the BRDF to be measured remotely without having to adjust either the incident or observation angle with respect to the target.

© 2008 Optical Society of America

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References

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  1. E. Baleine and A. Dogariu, “Variable-coherence tomography,” Opt. Lett. 29, 1233-1235 (2004).
    [CrossRef] [PubMed]
  2. E. Baleine and A. Dogariu, “Variable-coherence tomography for inverse scattering problems,” J. Opt. Soc. Am. A 21, 1917-1923 (2004).
    [CrossRef]
  3. W. H. Carter and E. Wolf, “Scattering from quasi-homogeneous media,” Opt. Commun. 67, 85-90 (1988).
    [CrossRef]
  4. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263-267 (2003).
    [CrossRef]
  5. E. Baleine and A. Dogariu, “Variable coherence scattering microscopy,” Phys. Rev. Lett. 95, 193904 (2005).
    [CrossRef] [PubMed]
  6. J. W. Goodman, Statistical Optics (Wiley Interscience, 2000).
  7. W. H. Carter and E. Wolf, “Coherence and radiometry with quasi-homogeneous planar sources,” J. Opt. Soc. Am. 67, 785-796 (1977).
    [CrossRef]
  8. D. G. Fischer and E. Wolf, “Theory of diffraction tomography for quasi-homogeneous random objects,” Opt. Commun. 133, 17-21 (1997).
    [CrossRef]
  9. D. F. J. James and E. Wolf, “Determination of the degree of coherence of light from spectroscopic measurements,” Opt. Commun. 145, 1-4 (1997).
    [CrossRef]
  10. D. Lara and C. Dainty, “Axially resolved complete Mueller matrix confocal microscopy,” Appl. Opt. 45, 1917-1930 (2006).
    [CrossRef] [PubMed]
  11. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999), Chap. 4.
  12. C.-T. Tai, Dyadic Green Functions in Electromagnetic Theory (IEEE, 1994).
  13. J. M. Harris, “The influence of random media on the propagation and depolarization of electromagnetic waves,” Ph.D. thesis (California Institute of Technology, 1980).
  14. A. Aiello and J. P. Woerdman, “Role of spatial coherence in polarization tomography,” Opt. Lett. 30, 1599-1601 (2005).
    [CrossRef] [PubMed]
  15. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).
  16. D. F. J. Arago and A. J. Fresnel, “On the action of rays of polarized light upon each other,” Ann. Chim. Phys. p. 288 (1819); [Translated in The Wave Theory of Light: Memoirs of Huygens, Young, and Fresnel, H.Crew, ed. (American Book Company, 1900), pp. 145-157].
  17. M. Mujat, A. Dogariu, and E. Wolf, “A law of interference of electromagnetic beams of any state of coherence and polarization and the Fresnel-Arago interference laws,” J. Opt. Soc. Am. A 21, 2414-2417 (2004).
    [CrossRef]
  18. F. E. Nicodemus, “Directional reflectance and emissivity of an opaque surface,” Appl. Opt. 4, 767-773 (1965).
    [CrossRef]
  19. D. H. Goldstein and D. B. Chenault, “Spectropolarimetric reflectometer,” Opt. Eng. 41, 1013-1020 (2002).
    [CrossRef]
  20. T. Wu and Y. Zhao, “The bidirectional polarized reflectance model of soil,” IEEE Trans. Geosci. Remote Sens. 43, 2854-2859 (2005).
    [CrossRef]

2006 (1)

2005 (3)

A. Aiello and J. P. Woerdman, “Role of spatial coherence in polarization tomography,” Opt. Lett. 30, 1599-1601 (2005).
[CrossRef] [PubMed]

E. Baleine and A. Dogariu, “Variable coherence scattering microscopy,” Phys. Rev. Lett. 95, 193904 (2005).
[CrossRef] [PubMed]

T. Wu and Y. Zhao, “The bidirectional polarized reflectance model of soil,” IEEE Trans. Geosci. Remote Sens. 43, 2854-2859 (2005).
[CrossRef]

2004 (3)

2003 (1)

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263-267 (2003).
[CrossRef]

2002 (1)

D. H. Goldstein and D. B. Chenault, “Spectropolarimetric reflectometer,” Opt. Eng. 41, 1013-1020 (2002).
[CrossRef]

1997 (2)

D. G. Fischer and E. Wolf, “Theory of diffraction tomography for quasi-homogeneous random objects,” Opt. Commun. 133, 17-21 (1997).
[CrossRef]

D. F. J. James and E. Wolf, “Determination of the degree of coherence of light from spectroscopic measurements,” Opt. Commun. 145, 1-4 (1997).
[CrossRef]

1988 (1)

W. H. Carter and E. Wolf, “Scattering from quasi-homogeneous media,” Opt. Commun. 67, 85-90 (1988).
[CrossRef]

1977 (1)

1965 (1)

1819 (1)

D. F. J. Arago and A. J. Fresnel, “On the action of rays of polarized light upon each other,” Ann. Chim. Phys. p. 288 (1819); [Translated in The Wave Theory of Light: Memoirs of Huygens, Young, and Fresnel, H.Crew, ed. (American Book Company, 1900), pp. 145-157].

Aiello, A.

Arago, D. F. J.

D. F. J. Arago and A. J. Fresnel, “On the action of rays of polarized light upon each other,” Ann. Chim. Phys. p. 288 (1819); [Translated in The Wave Theory of Light: Memoirs of Huygens, Young, and Fresnel, H.Crew, ed. (American Book Company, 1900), pp. 145-157].

Azzam, R. M. A.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

Baleine, E.

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

Carter, W. H.

W. H. Carter and E. Wolf, “Scattering from quasi-homogeneous media,” Opt. Commun. 67, 85-90 (1988).
[CrossRef]

W. H. Carter and E. Wolf, “Coherence and radiometry with quasi-homogeneous planar sources,” J. Opt. Soc. Am. 67, 785-796 (1977).
[CrossRef]

Chenault, D. B.

D. H. Goldstein and D. B. Chenault, “Spectropolarimetric reflectometer,” Opt. Eng. 41, 1013-1020 (2002).
[CrossRef]

Dainty, C.

Dogariu, A.

Fischer, D. G.

D. G. Fischer and E. Wolf, “Theory of diffraction tomography for quasi-homogeneous random objects,” Opt. Commun. 133, 17-21 (1997).
[CrossRef]

Fresnel, A. J.

D. F. J. Arago and A. J. Fresnel, “On the action of rays of polarized light upon each other,” Ann. Chim. Phys. p. 288 (1819); [Translated in The Wave Theory of Light: Memoirs of Huygens, Young, and Fresnel, H.Crew, ed. (American Book Company, 1900), pp. 145-157].

Goldstein, D. H.

D. H. Goldstein and D. B. Chenault, “Spectropolarimetric reflectometer,” Opt. Eng. 41, 1013-1020 (2002).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley Interscience, 2000).

Harris, J. M.

J. M. Harris, “The influence of random media on the propagation and depolarization of electromagnetic waves,” Ph.D. thesis (California Institute of Technology, 1980).

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999), Chap. 4.

James, D. F. J.

D. F. J. James and E. Wolf, “Determination of the degree of coherence of light from spectroscopic measurements,” Opt. Commun. 145, 1-4 (1997).
[CrossRef]

Lara, D.

Mujat, M.

Nicodemus, F. E.

Tai, C.-T.

C.-T. Tai, Dyadic Green Functions in Electromagnetic Theory (IEEE, 1994).

Woerdman, J. P.

Wolf, E.

M. Mujat, A. Dogariu, and E. Wolf, “A law of interference of electromagnetic beams of any state of coherence and polarization and the Fresnel-Arago interference laws,” J. Opt. Soc. Am. A 21, 2414-2417 (2004).
[CrossRef]

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263-267 (2003).
[CrossRef]

D. G. Fischer and E. Wolf, “Theory of diffraction tomography for quasi-homogeneous random objects,” Opt. Commun. 133, 17-21 (1997).
[CrossRef]

D. F. J. James and E. Wolf, “Determination of the degree of coherence of light from spectroscopic measurements,” Opt. Commun. 145, 1-4 (1997).
[CrossRef]

W. H. Carter and E. Wolf, “Scattering from quasi-homogeneous media,” Opt. Commun. 67, 85-90 (1988).
[CrossRef]

W. H. Carter and E. Wolf, “Coherence and radiometry with quasi-homogeneous planar sources,” J. Opt. Soc. Am. 67, 785-796 (1977).
[CrossRef]

Wu, T.

T. Wu and Y. Zhao, “The bidirectional polarized reflectance model of soil,” IEEE Trans. Geosci. Remote Sens. 43, 2854-2859 (2005).
[CrossRef]

Zhao, Y.

T. Wu and Y. Zhao, “The bidirectional polarized reflectance model of soil,” IEEE Trans. Geosci. Remote Sens. 43, 2854-2859 (2005).
[CrossRef]

Ann. Chim. Phys. (1)

D. F. J. Arago and A. J. Fresnel, “On the action of rays of polarized light upon each other,” Ann. Chim. Phys. p. 288 (1819); [Translated in The Wave Theory of Light: Memoirs of Huygens, Young, and Fresnel, H.Crew, ed. (American Book Company, 1900), pp. 145-157].

Appl. Opt. (2)

IEEE Trans. Geosci. Remote Sens. (1)

T. Wu and Y. Zhao, “The bidirectional polarized reflectance model of soil,” IEEE Trans. Geosci. Remote Sens. 43, 2854-2859 (2005).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (3)

W. H. Carter and E. Wolf, “Scattering from quasi-homogeneous media,” Opt. Commun. 67, 85-90 (1988).
[CrossRef]

D. G. Fischer and E. Wolf, “Theory of diffraction tomography for quasi-homogeneous random objects,” Opt. Commun. 133, 17-21 (1997).
[CrossRef]

D. F. J. James and E. Wolf, “Determination of the degree of coherence of light from spectroscopic measurements,” Opt. Commun. 145, 1-4 (1997).
[CrossRef]

Opt. Eng. (1)

D. H. Goldstein and D. B. Chenault, “Spectropolarimetric reflectometer,” Opt. Eng. 41, 1013-1020 (2002).
[CrossRef]

Opt. Lett. (2)

Phys. Lett. A (1)

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263-267 (2003).
[CrossRef]

Phys. Rev. Lett. (1)

E. Baleine and A. Dogariu, “Variable coherence scattering microscopy,” Phys. Rev. Lett. 95, 193904 (2005).
[CrossRef] [PubMed]

Other (5)

J. W. Goodman, Statistical Optics (Wiley Interscience, 2000).

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999), Chap. 4.

C.-T. Tai, Dyadic Green Functions in Electromagnetic Theory (IEEE, 1994).

J. M. Harris, “The influence of random media on the propagation and depolarization of electromagnetic waves,” Ph.D. thesis (California Institute of Technology, 1980).

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

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Figures (2)

Fig. 1
Fig. 1

Polarimetric interferometer for creating a beam with coherence between the orthogonal polarization components. A detailed analysis of this interferometer will be considered elsewhere, but the concept helps illustrate the principles in the text.

Fig. 2
Fig. 2

Geometry for computing the BRDF of a surface. In order to fully characterize the BRDF, the full range of scattering angles ( θ , ϕ ) must be scanned for all incident angles ( θ , ϕ ) .

Equations (42)

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V ( i ) ( r , t ) = U ( i ) ( r ) exp ( i ω t ) ,
U ( s ) ( r , ω ) = V F ( r , ω ) U ( i ) ( r , ω ) exp ( i k r r ) r r d r .
F ( r , ω ) = ( 4 π ) 1 k 2 ( n 2 ( r , ω ) 1 ) ,
U ( s ) ( r , ω ) exp ( i k r ) r V F ( r , ω ) U ( i ) ( r , ω ) exp ( i k s ̂ r ) d r
U ̃ ( r ) = U ( r , ω ) .
W ̃ ( α ) ( r 1 , r 2 ) = U ( α ) ( r 1 ) U ̃ ( α ) ( r 2 ) * U ,
C ̃ F ( r 1 , r 2 ) = F ̃ ( r 1 ) F ̃ ( r 2 ) * F ,
C ̃ F ( r 1 , r 2 ) S ̃ f ( r ¯ ) μ ̃ F ( Δ r ) ,
W ̃ ( i ) ( r 1 , r 2 ) = I ̃ ( i ) ( r ¯ ) μ ̃ ( i ) ( Δ r ) ,
F ̃ U ̃ ( α ) F ̃ * U ̃ ( α ) * = F ̃ U ̃ ( α ) F ̃ * U ̃ ( α ) * F U
= F ̃ F ̃ * F U ̃ ( α ) U ̃ ( α ) * U
= C ̃ F W ̃ ( α ) .
W ̃ ( s ) ( r 1 , r 2 ) = V V S ̃ F ( r ¯ ) μ ̃ F ( Δ r ) I ̃ ( i ) ( r ¯ ) μ ̃ ( i ) ( Δ r ) exp ( i ( s ̂ 1 r 1 s ̂ 2 r 2 ) ) d r 1 d r 2 .
I ̃ ( s ) ( r s ̂ ) = W ̃ ( s ) ( r s ̂ , r s ̂ ) = 1 r 2 V V S F ( r ¯ ) μ ̃ F ( Δ r ) I ̃ ( i ) ( r ¯ ) μ ̃ ( i ) ( Δ r ) exp ( i k s ̂ Δ r ) d r 1 d r 2 .
μ ̃ ( i ) ( Δ r ) exp ( i k Δ z ) δ ( Δ r ) + m 2 exp { i [ ϕ ( r 0 ) k Δ z ] } δ ( Δ r + r 0 ) + m 2 exp { i [ ϕ ( r 0 ) k Δ z ] } δ ( Δ r r 0 ) .
I ̃ ( s ) ( r s ̂ ; r 0 ) = 1 r 2 V S F ( r 1 ) I ( i ) ( r 1 ) [ e i k Δ z μ ̃ F ( 0 ) + m 2 e i k ( ϕ k Δ z ) μ ̃ F ( r 0 ) + m 2 e i k ( ϕ k Δ z ) μ ̃ F ( r 0 ) ] d r 1 .
E ̃ ( α ) ( r ) = [ E ̃ x ( α ) ( r ) E ̃ y ( α ) ( r ) ] .
W ͇ ̃ ( α ) ( r 1 , r 2 ) = E ̃ ( α ) ( r 1 ) E ̃ ( α ) ( r 2 ) U ,
F ̃ ( r ) = k 2 4 π ( n ̃ 2 ( r ) 1 ) = k 2 4 π χ ̃ e ( r ) ,
F ͇ ̃ ( r ) = k 2 4 π χ ͇ ̃ e ( r ) .
E ( s ) ( r , ω ) = V F ͇ ( r , ω ) E ( i ) ( r , ω ) exp ( i k r r ) r r d r .
W ͇ ̃ ( s ) ( r 1 , r 2 ) = exp ( i k ( r 1 r 2 ) ) r 1 r 2 V V F ͇ ̃ ( r 1 ) E ̃ ( i ) ( r 1 ) E ̃ ( i ) ( r 2 ) U F ͇ ̃ ( r 2 ) F exp ( i k s ̂ Δ r ) d r 1 d r 2 .
W ͇ ̃ ( s ) ( r 1 , r 2 ) = exp ( i k ( r 1 r 2 ) ) r 1 r 2 V V F ͇ ̃ ( r 1 ) W ͇ ̃ ( i ) ( r 1 , r 2 ) F ͇ ̃ ( r 2 ) F exp ( i k s ̂ Δ r ) d r 1 d r 2 .
J ͇ ̃ ( s ) ( r s ̂ ) = W ͇ ̃ ( s ) ( r s ̂ , r s ̂ ) = 1 r 2 V V F ͇ ̃ ( r 1 ) W ͇ ̃ ( i ) ( r 1 , r 2 ) F ͇ ̃ ( r 2 ) F exp ( i k s ̂ Δ r ) d r 1 d r 2 .
C ͇ ̃ F ( r 1 , r 2 ) = F ͇ ̃ ( r 1 ) F ͇ ̃ ( r 2 ) F
W ̃ ( r 1 , r 2 ) = I ̃ ( r ¯ ) μ ( Δ r ) = I ̃ ( r ¯ ) μ ( Δ r ) I ̃ ( r ¯ ) .
W ͇ ̃ ( r 1 , r 2 ) = I ͇ ̃ 1 2 ( r ¯ ) μ ͇ ̃ ( Δ r ) I ͇ 1 2 ( r ¯ ) ,
I ͇ ̃ 1 2 ( r ) = [ I ̃ x ( r ) 0 0 I ̃ y ( r ) ]
μ ̃ i j ( Δ r ) = E ̃ i ( Δ r 2 ) I ̃ i ( 0 ) E ̃ j ( Δ r 2 ) * I ̃ j ( 0 ) .
W ͇ ̃ pol ( i ) ( r 1 , r 2 ) = I ̃ pol ( i ) ( r ¯ ) μ ( i ) ( Δ r ) p ̂ p ̂ ,
F ͇ = R ͇ p 1 F ͇ p R ͇ p ,
R ͇ p J ͇ ̃ ( r s ̂ ) R ͇ p 1 = 1 r 2 V V [ χ ̃ p p ( r 1 ) χ ̃ p p * ( r 2 ) χ ̃ p p ( r 1 ) χ ̃ q p * ( r 2 ) χ ̃ q p ( r 1 ) χ ̃ p p * ( r 2 ) χ ̃ q p ( r 1 χ ̃ q p * ( r 2 ) ) ] × I ̃ pol ( i ) ( r ¯ ) μ ( i ) ( Δ r ) exp ( i k s ̂ Δ r ) d r 12 d r 2 .
C ̃ i j , i j ( r 1 , r 2 ) = χ ̃ i j ( r 1 ) χ ̃ i j * ( r 2 ) = S ̃ F , i j , i j ( r ¯ ) μ F , i j , i j ( Δ r ) ,
U ̃ ( r 1 ) U ̃ * ( r 2 ) U I ( r ¯ ) δ ( Δ r ) ,
E out = 1 2 [ U ( r ) U ( r r 0 ) ] ,
W ͇ ̃ ( r 1 , r 2 ) = I ( r ¯ ) 4 [ δ ( r 1 r 2 ) δ ( r 1 r 2 + r 0 ) δ ( r 1 r 2 r 0 ) δ ( r 1 r 2 ) ] = I ( r ¯ ) 4 { δ ( Δ r ) [ 1 0 0 1 ] + δ ( Δ r r 0 ) [ 0 1 0 0 ] + δ ( Δ r + r 0 ) [ 0 0 1 0 ] } .
J ͇ ̃ ( s ) ( r s ̂ ; r 0 ) = 1 r 2 V [ C ͇ ̃ F ( r 1 , r 1 ) + R ͇ ̃ F ( r 1 , r 0 ) e i k s ̂ r 0 + R ͇ ̃ F ( r 1 , r 0 ) e i k s ̂ r 0 ] d r 1 ,
R ͇ ̃ F = exp ( i k s ̂ r 0 ) [ S ̃ F , 11 , 12 ( r 1 ) μ F , 11 , 12 ( r 0 ) S ̃ F , 11 , 22 ( r 1 ) μ F , 11 , 22 ( r 0 ) S ̃ F , 21 , 12 ( r 1 ) μ F , 21 , 12 ( r 0 ) S ̃ F , 21 , 22 ( r 1 ) μ F , 21 , 22 ( r 0 ) ] .
U ( i ) ( r ) = U 0 exp ( i k k ̂ i r ) .
U ( s ) ( θ , ϕ ) = U s ( δ ( k ̂ k ̂ s ) + A F ( x , y ) U ( i ) ( x , y ) exp ( i k r r ) r r d x d y ) .
I ( k ̂ ) = Ω U ( s ) ( θ , ϕ ) 2 d θ d ϕ
I ( r k ̂ ) = 1 r 2 A C ̃ F ( r 1 , r 2 ) W ̃ ( r 1 , r 2 ) d r 1 d r 2

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