Abstract

We analyze the effect of partial spatial coherence on the scattering of light by an arbitrary particle. We extend the definition of the extinction cross section to spatially partially coherent fields. We then discuss the effect of the partial coherence on the extinction scattering cross section by introducing the Wigner transform. It is shown that for rotationally invariant scatterers, the extinction cross section does not depend on the coherence of the incident field. The effect of partial coherence on the angular behavior of the scattered intensity is also discussed in the framework of the Wigner transform. The implications for practical applications are considered.

© 2003 Optical Society of America

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References

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  1. H. C. van de Hulst, Light Scattering by Small Particles (Dover, Toronto, Ontario, Canada, 1981).
  2. C. F. Bohren, D. R. Huffmann, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  3. M. Nieto-Vesperinas, G. Ross, M. A. Fiddy, “The optical theorems: a new interpretation for partially coherent light,” Optik (Stuttgart) 55, 165–171 (1980).
  4. P. S. Carney, E. Wolf, G. S. Agarwal, “Statistical generalizations of the optical cross-section theorem with application to inverse scattering,” J. Opt. Soc. Am. A 14, 3366–3371 (1997).
    [CrossRef]
  5. P. S. Carney, E. Wolf, G. S. Agarwal, “Diffraction tomography using power extinction measurements,” J. Opt. Soc. Am. A 16, 2643–2648 (1999).
    [CrossRef]
  6. P. S. Carney, E. Wolf, “An energy theorem for scattering of partially coherent beams,” Opt. Commun. 155, 1–6 (1998). Note that the conclusions of this paper appear to contradict our conclusions. This is due to a sign error in Eq. (3.3), in which the signs between (s1⊥+s2⊥) and (s1⊥-s2⊥) have been exchanged.
    [CrossRef]
  7. J. Jannson, T. Jannson, E. Wolf, “Spatial coherence discrimination in scattering,” Opt. Lett. 13, 1060–1062 (1988).
    [CrossRef] [PubMed]
  8. F. Gori, C. Palma, M. Santarsiero, “A scattering experiment with partially coherent light,” Opt. Commun. 74, 353–356 (1990).
    [CrossRef]
  9. D. Cabaret, S. Rossano, C. Brouder, “Mie scattering of a partially coherent beam,” Opt. Commun. 150, 239–250 (1998).
    [CrossRef]
  10. M. Dusek, “Diffraction grating illuminated by partially coherent beam,” Opt. Commun. 111, 203–208 (1994).
    [CrossRef]
  11. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995).
  12. S. Ponomarenko, J. J. Greffet, E. Wolf, “The diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1–8 (2002).
    [CrossRef]
  13. G. Gbur, E. Wolf, “Spreading of coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592–1598 (2002).
    [CrossRef]
  14. S. Chandrasekhar, Radiative Transfer Theory (Dover, New York, 1960).
  15. L. Ryzhik, G. Papanicolaou, J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327–370 (1996).
    [CrossRef]
  16. S. M. Rytov, Y. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics, Vol. 4: Wave Propagation through Random Media (Springer-Verlag, Berlin, 1989).

2002 (2)

S. Ponomarenko, J. J. Greffet, E. Wolf, “The diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1–8 (2002).
[CrossRef]

G. Gbur, E. Wolf, “Spreading of coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592–1598 (2002).
[CrossRef]

1999 (1)

1998 (2)

P. S. Carney, E. Wolf, “An energy theorem for scattering of partially coherent beams,” Opt. Commun. 155, 1–6 (1998). Note that the conclusions of this paper appear to contradict our conclusions. This is due to a sign error in Eq. (3.3), in which the signs between (s1⊥+s2⊥) and (s1⊥-s2⊥) have been exchanged.
[CrossRef]

D. Cabaret, S. Rossano, C. Brouder, “Mie scattering of a partially coherent beam,” Opt. Commun. 150, 239–250 (1998).
[CrossRef]

1997 (1)

1996 (1)

L. Ryzhik, G. Papanicolaou, J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327–370 (1996).
[CrossRef]

1994 (1)

M. Dusek, “Diffraction grating illuminated by partially coherent beam,” Opt. Commun. 111, 203–208 (1994).
[CrossRef]

1990 (1)

F. Gori, C. Palma, M. Santarsiero, “A scattering experiment with partially coherent light,” Opt. Commun. 74, 353–356 (1990).
[CrossRef]

1988 (1)

1980 (1)

M. Nieto-Vesperinas, G. Ross, M. A. Fiddy, “The optical theorems: a new interpretation for partially coherent light,” Optik (Stuttgart) 55, 165–171 (1980).

Agarwal, G. S.

Bohren, C. F.

C. F. Bohren, D. R. Huffmann, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Brouder, C.

D. Cabaret, S. Rossano, C. Brouder, “Mie scattering of a partially coherent beam,” Opt. Commun. 150, 239–250 (1998).
[CrossRef]

Cabaret, D.

D. Cabaret, S. Rossano, C. Brouder, “Mie scattering of a partially coherent beam,” Opt. Commun. 150, 239–250 (1998).
[CrossRef]

Carney, P. S.

P. S. Carney, E. Wolf, G. S. Agarwal, “Diffraction tomography using power extinction measurements,” J. Opt. Soc. Am. A 16, 2643–2648 (1999).
[CrossRef]

P. S. Carney, E. Wolf, “An energy theorem for scattering of partially coherent beams,” Opt. Commun. 155, 1–6 (1998). Note that the conclusions of this paper appear to contradict our conclusions. This is due to a sign error in Eq. (3.3), in which the signs between (s1⊥+s2⊥) and (s1⊥-s2⊥) have been exchanged.
[CrossRef]

P. S. Carney, E. Wolf, G. S. Agarwal, “Statistical generalizations of the optical cross-section theorem with application to inverse scattering,” J. Opt. Soc. Am. A 14, 3366–3371 (1997).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer Theory (Dover, New York, 1960).

Dusek, M.

M. Dusek, “Diffraction grating illuminated by partially coherent beam,” Opt. Commun. 111, 203–208 (1994).
[CrossRef]

Fiddy, M. A.

M. Nieto-Vesperinas, G. Ross, M. A. Fiddy, “The optical theorems: a new interpretation for partially coherent light,” Optik (Stuttgart) 55, 165–171 (1980).

Gbur, G.

Gori, F.

F. Gori, C. Palma, M. Santarsiero, “A scattering experiment with partially coherent light,” Opt. Commun. 74, 353–356 (1990).
[CrossRef]

Greffet, J. J.

S. Ponomarenko, J. J. Greffet, E. Wolf, “The diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1–8 (2002).
[CrossRef]

Huffmann, D. R.

C. F. Bohren, D. R. Huffmann, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Jannson, J.

Jannson, T.

Keller, J. B.

L. Ryzhik, G. Papanicolaou, J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327–370 (1996).
[CrossRef]

Kravtsov, Y. A.

S. M. Rytov, Y. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics, Vol. 4: Wave Propagation through Random Media (Springer-Verlag, Berlin, 1989).

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995).

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, G. Ross, M. A. Fiddy, “The optical theorems: a new interpretation for partially coherent light,” Optik (Stuttgart) 55, 165–171 (1980).

Palma, C.

F. Gori, C. Palma, M. Santarsiero, “A scattering experiment with partially coherent light,” Opt. Commun. 74, 353–356 (1990).
[CrossRef]

Papanicolaou, G.

L. Ryzhik, G. Papanicolaou, J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327–370 (1996).
[CrossRef]

Ponomarenko, S.

S. Ponomarenko, J. J. Greffet, E. Wolf, “The diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1–8 (2002).
[CrossRef]

Ross, G.

M. Nieto-Vesperinas, G. Ross, M. A. Fiddy, “The optical theorems: a new interpretation for partially coherent light,” Optik (Stuttgart) 55, 165–171 (1980).

Rossano, S.

D. Cabaret, S. Rossano, C. Brouder, “Mie scattering of a partially coherent beam,” Opt. Commun. 150, 239–250 (1998).
[CrossRef]

Rytov, S. M.

S. M. Rytov, Y. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics, Vol. 4: Wave Propagation through Random Media (Springer-Verlag, Berlin, 1989).

Ryzhik, L.

L. Ryzhik, G. Papanicolaou, J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327–370 (1996).
[CrossRef]

Santarsiero, M.

F. Gori, C. Palma, M. Santarsiero, “A scattering experiment with partially coherent light,” Opt. Commun. 74, 353–356 (1990).
[CrossRef]

Tatarskii, V. I.

S. M. Rytov, Y. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics, Vol. 4: Wave Propagation through Random Media (Springer-Verlag, Berlin, 1989).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, Toronto, Ontario, Canada, 1981).

Wolf, E.

S. Ponomarenko, J. J. Greffet, E. Wolf, “The diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1–8 (2002).
[CrossRef]

G. Gbur, E. Wolf, “Spreading of coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592–1598 (2002).
[CrossRef]

P. S. Carney, E. Wolf, G. S. Agarwal, “Diffraction tomography using power extinction measurements,” J. Opt. Soc. Am. A 16, 2643–2648 (1999).
[CrossRef]

P. S. Carney, E. Wolf, “An energy theorem for scattering of partially coherent beams,” Opt. Commun. 155, 1–6 (1998). Note that the conclusions of this paper appear to contradict our conclusions. This is due to a sign error in Eq. (3.3), in which the signs between (s1⊥+s2⊥) and (s1⊥-s2⊥) have been exchanged.
[CrossRef]

P. S. Carney, E. Wolf, G. S. Agarwal, “Statistical generalizations of the optical cross-section theorem with application to inverse scattering,” J. Opt. Soc. Am. A 14, 3366–3371 (1997).
[CrossRef]

J. Jannson, T. Jannson, E. Wolf, “Spatial coherence discrimination in scattering,” Opt. Lett. 13, 1060–1062 (1988).
[CrossRef] [PubMed]

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995).

J. Opt. Soc. Am. A (3)

Opt. Commun. (5)

F. Gori, C. Palma, M. Santarsiero, “A scattering experiment with partially coherent light,” Opt. Commun. 74, 353–356 (1990).
[CrossRef]

D. Cabaret, S. Rossano, C. Brouder, “Mie scattering of a partially coherent beam,” Opt. Commun. 150, 239–250 (1998).
[CrossRef]

M. Dusek, “Diffraction grating illuminated by partially coherent beam,” Opt. Commun. 111, 203–208 (1994).
[CrossRef]

S. Ponomarenko, J. J. Greffet, E. Wolf, “The diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1–8 (2002).
[CrossRef]

P. S. Carney, E. Wolf, “An energy theorem for scattering of partially coherent beams,” Opt. Commun. 155, 1–6 (1998). Note that the conclusions of this paper appear to contradict our conclusions. This is due to a sign error in Eq. (3.3), in which the signs between (s1⊥+s2⊥) and (s1⊥-s2⊥) have been exchanged.
[CrossRef]

Opt. Lett. (1)

Optik (Stuttgart) (1)

M. Nieto-Vesperinas, G. Ross, M. A. Fiddy, “The optical theorems: a new interpretation for partially coherent light,” Optik (Stuttgart) 55, 165–171 (1980).

Wave Motion (1)

L. Ryzhik, G. Papanicolaou, J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327–370 (1996).
[CrossRef]

Other (5)

S. M. Rytov, Y. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics, Vol. 4: Wave Propagation through Random Media (Springer-Verlag, Berlin, 1989).

S. Chandrasekhar, Radiative Transfer Theory (Dover, New York, 1960).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, Toronto, Ontario, Canada, 1981).

C. F. Bohren, D. R. Huffmann, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995).

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Equations (22)

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Ψ(i)(r, t)=Re[ψ(i)(r)exp(-iωt)]
ψ(i)(r)=a exp(iksi·r),
Ψ(s)(r, t)=Re[ψ(s)(r)exp(-iωt)].
Ψ(s)(r, t)a exp(ikr)krf(ss, si; s0, ω).
Pe(s0, si, ω)=4πk2|a|2 Im[f(si, si; s0, ω)],
σe(s0, si, ω)=4πk2Im[f(si, si; s0, ω)].
ψ(i)(r)=4πa(i)(s)exp(iks·r)|sz|dΩ,
A(i)(s, s)=a(i)*(s)a(i)(s).
Pe(s0, ω)=4πk24π4π Im[A(i)(s, s)f(s, s; s0, ω)]×|sz|dΩ|sz|dΩ.
I(0)=|ψ(i)(0)|2=4π4πA(i)(s, s)|sz|dΩ|sz|dΩ.
σe(ω)=4πk2×4π4π Im[A(i)(s, s)f(s, s; s0, ω)]|sz|dΩ|sz|dΩ4π4πA(i)(s, s)|sz|dΩ|sz|dΩ.
=A(i)(s, s)exp[-ik(s·r-s·r)]|sz||sz|dΩdΩ.
ψ(i)*R+ρ2ψ(i)R-ρ2=I(i)(R, S)exp(ikS·ρ)|Sz|dΩ,
I(i)(R, S)=A(i)S-S2, S+S2exp(-ikS·R)|Sz|dΩ,
4π4π ImA(i)S-S2, S+S2×fS-S2, S+S2; s0, ω|Sz|dΩ|Sz|dΩ.
λ/σIλ/a,
4π Imf(S, S; s0, ω)|Sz|dΩ4πA(i)S-S2, S+S2|Sz|dΩ.
I(i)(0, S)=4πA(i)S-S2, S+S2|Sz|dΩ.
σe(s0, ω)=4πk24πI(i)(0, S)Im[f(S, S; s0, ω)]|Sz|dΩ4πI(0, S)|Sz|dΩ.
|Ψs(r)|21k2r24π|sz1|dΩ14π|sz2|dΩ2×f*(s, s1; s0, ω)f(s, s2; s0, ω)×ai*(s1)ai(s2),
|Ψs(r)|21k2r24π|Sz|dΩ4π|Sz|dΩfs, S+S2; s0, ωfs, S-S2; s0, ω×ai*S+S2aiS-S2.
|Ψs(r)|21k2r24π2πk2|f(s, S; s0, ω)|2I(i)×(0, S)|Sz|dΩ.

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