Abstract

The scattering of a partially coherent beam by a deterministic, spherical scatterer is studied. In particular, the Mie scattering by a Gaussian Schell-model beam is analyzed. Expressions are derived for (a) the extinguished power, (b) the radiant intensity of the scattered field, and (c) the encircled energy in the far field. It is found that the radiant intensity and the encircled energy in the far field depend on the degree of coherence of the incident beam, whereas the extinguished power does not.

© 2012 Optical Society of America

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References

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  1. G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. IV, Folge 25, 377–445 (1908).
    [CrossRef]
  2. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957). See Chap. 9.
  3. M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University, 1999).
  4. H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge University, 1992). See Chap. 5.
    [CrossRef]
  5. W. T. Grandy, Jr., Scattering of Waves from Large Spheres (Cambridge University, 1992). See Chap. 3.
  6. P. S. Carney, E. Wolf, and 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]
  7. P. S. Carney and E. Wolf, “An energy theorem for scattering of partially coherent beams,” Opt. Commun. 155, 1–6 (1998).
    [CrossRef]
  8. D. Cabaret, S. Rossano, and C. Brouder, “Mie scattering of a partially coherent beam,” Opt. Commun. 150, 239–250 (1998).
    [CrossRef]
  9. J. J. Greffet, M. De La Cruz-Gutierrez, P. V. Ignatovich, and A. Radunsky, “Influence of spatial coherence on scattering by a particle,” J. Opt. Soc. Am. A 20, 2315–2320 (2003).
    [CrossRef]
  10. T. D. Visser, D. G. Fischer, and E. Wolf, “Scattering of light from quasi-homogeneous sources by quasi-homogeneous media,” J. Opt. Soc. Am. A 23, 1631–1638 (2006).
    [CrossRef]
  11. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007). See Chap. 6.
  12. T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104, 173902 (2010).
    [CrossRef] [PubMed]
  13. G. Gbur and T. D. Visser, “The structure of partially coherent fields,” in Vol.  55 of Progress in Optics, E.Wolf, ed. (Elsevier, 2010), pp. 285–341.
    [CrossRef]
  14. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  15. C. J. Joachain, Quantum Collision Theory, 3rd ed. (Elsevier, 1987).
  16. D. Zwillinger, Handbook of Integration (Jones and Bartlett, 1992), Chap. 8.
  17. F. Gori, C. Palma, and M. Santarsiero, “A scattering experiment with partially coherent light,” Opt. Commun. 74, 353–356 (1990).
    [CrossRef]
  18. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, 2nd ed. (McGraw, 1968). See p. 852.
  19. M. W. Kowarz and E. Wolf, “Conservation laws for partially coherent free fields,” J. Opt. Soc. Am. A 10, 88–94 (1993).
    [CrossRef]

2010 (1)

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104, 173902 (2010).
[CrossRef] [PubMed]

2006 (1)

2003 (1)

1998 (2)

P. S. Carney and E. Wolf, “An energy theorem for scattering of partially coherent beams,” Opt. Commun. 155, 1–6 (1998).
[CrossRef]

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

1997 (1)

1993 (1)

1990 (1)

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

1908 (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. IV, Folge 25, 377–445 (1908).
[CrossRef]

Agarwal, G. S.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University, 1999).

Brouder, C.

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

Cabaret, D.

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

Carney, P. S.

De La Cruz-Gutierrez, M.

Fischer, D. G.

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104, 173902 (2010).
[CrossRef] [PubMed]

T. D. Visser, D. G. Fischer, and E. Wolf, “Scattering of light from quasi-homogeneous sources by quasi-homogeneous media,” J. Opt. Soc. Am. A 23, 1631–1638 (2006).
[CrossRef]

Gbur, G.

G. Gbur and T. D. Visser, “The structure of partially coherent fields,” in Vol.  55 of Progress in Optics, E.Wolf, ed. (Elsevier, 2010), pp. 285–341.
[CrossRef]

Gori, F.

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

Grandy, W. T.

W. T. Grandy, Jr., Scattering of Waves from Large Spheres (Cambridge University, 1992). See Chap. 3.

Greffet, J. J.

Ignatovich, P. V.

Joachain, C. J.

C. J. Joachain, Quantum Collision Theory, 3rd ed. (Elsevier, 1987).

Korn, G. A.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, 2nd ed. (McGraw, 1968). See p. 852.

Korn, T. M.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, 2nd ed. (McGraw, 1968). See p. 852.

Kowarz, M. W.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Mie, G.

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. IV, Folge 25, 377–445 (1908).
[CrossRef]

Nussenzveig, H. M.

H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge University, 1992). See Chap. 5.
[CrossRef]

Palma, C.

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

Radunsky, A.

Rossano, S.

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

Santarsiero, M.

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

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957). See Chap. 9.

van Dijk, T.

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104, 173902 (2010).
[CrossRef] [PubMed]

Visser, T. D.

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104, 173902 (2010).
[CrossRef] [PubMed]

T. D. Visser, D. G. Fischer, and E. Wolf, “Scattering of light from quasi-homogeneous sources by quasi-homogeneous media,” J. Opt. Soc. Am. A 23, 1631–1638 (2006).
[CrossRef]

G. Gbur and T. D. Visser, “The structure of partially coherent fields,” in Vol.  55 of Progress in Optics, E.Wolf, ed. (Elsevier, 2010), pp. 285–341.
[CrossRef]

Wolf, E.

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104, 173902 (2010).
[CrossRef] [PubMed]

T. D. Visser, D. G. Fischer, and E. Wolf, “Scattering of light from quasi-homogeneous sources by quasi-homogeneous media,” J. Opt. Soc. Am. A 23, 1631–1638 (2006).
[CrossRef]

P. S. Carney and E. Wolf, “An energy theorem for scattering of partially coherent beams,” Opt. Commun. 155, 1–6 (1998).
[CrossRef]

P. S. Carney, E. Wolf, and 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]

M. W. Kowarz and E. Wolf, “Conservation laws for partially coherent free fields,” J. Opt. Soc. Am. A 10, 88–94 (1993).
[CrossRef]

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007). See Chap. 6.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University, 1999).

Zwillinger, D.

D. Zwillinger, Handbook of Integration (Jones and Bartlett, 1992), Chap. 8.

Ann. Phys. (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. IV, Folge 25, 377–445 (1908).
[CrossRef]

J. Opt. Soc. Am. A (4)

Opt. Commun. (3)

P. S. Carney and E. Wolf, “An energy theorem for scattering of partially coherent beams,” Opt. Commun. 155, 1–6 (1998).
[CrossRef]

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

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

Phys. Rev. Lett. (1)

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104, 173902 (2010).
[CrossRef] [PubMed]

Other (10)

G. Gbur and T. D. Visser, “The structure of partially coherent fields,” in Vol.  55 of Progress in Optics, E.Wolf, ed. (Elsevier, 2010), pp. 285–341.
[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

C. J. Joachain, Quantum Collision Theory, 3rd ed. (Elsevier, 1987).

D. Zwillinger, Handbook of Integration (Jones and Bartlett, 1992), Chap. 8.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957). See Chap. 9.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University, 1999).

H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge University, 1992). See Chap. 5.
[CrossRef]

W. T. Grandy, Jr., Scattering of Waves from Large Spheres (Cambridge University, 1992). See Chap. 3.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, 2nd ed. (McGraw, 1968). See p. 852.

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007). See Chap. 6.

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Figures (11)

Fig. 1
Fig. 1

Illustrating the notation.

Fig. 2
Fig. 2

Normalized angular distribution of the radiant intensity generated by scattering a partially coherent beam by a sphere, for various values of the transverse coherence length σ μ . In this example, the sphere radius a = 1 λ , and the refractive index n = 2 .

Fig. 3
Fig. 3

Normalized angular distribution of the radiant intensity generated by scattering a partially coherent beam by a sphere, for various values of the transverse coherence length σ μ . In this example, the sphere radius a = 2 λ , and the refractive index n = 2 .

Fig. 4
Fig. 4

Normalized angular distribution of the radiant intensity generated by scattering a partially coherent beam by a sphere, for various values of the transverse coherence length σ μ . In this example, the sphere radius a = 4 λ , and the refractive index n = 2 .

Fig. 5
Fig. 5

Normalized angular distribution of the radiant intensity generated by scattering a partially coherent beam by a sphere, shown on a logarithmic scale, for various values of the transverse coherence length σ μ . In this example, the sphere radius a = 4 λ , and the refractive index n = 2 .

Fig. 6
Fig. 6

Normalized intercepted power as a function of the angle subtended by the detector for various values of the transverse coherence length σ μ . In this example, the sphere radius a = 1 λ , and the refractive index n = 2 .

Fig. 7
Fig. 7

Normalized intercepted power as a function of the angle subtended by the detector for various values of the transverse coherence length σ μ . In this example, the sphere radius a = 2 λ , and the refractive index n = 2 .

Fig. 8
Fig. 8

Normalized intercepted power as a function of the angle subtended by the detector for various values of the transverse coherence length σ μ . In this example, the sphere radius a = 4 λ , and the refractive index n = 2 .

Fig. 9
Fig. 9

Normalized intercepted power as a function of the transverse coherence length σ μ for several values of the angle subtended by the detector. In this example, the sphere radius a = 1 λ , and the refractive index n = 2 .

Fig. 10
Fig. 10

Normalized intercepted power as a function of the transverse coherence length σ μ for several values of the angle subtended by the detector. In this example, the sphere radius a = 2 λ , and the refractive index n = 2 .

Fig. 11
Fig. 11

Normalized intercepted power as a function of the transverse coherence length σ μ for several values of the angle subtended by the detector. In this example, the sphere radius a = 4 λ , and the refractive index n = 2 .

Equations (47)

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V ( i ) ( r , t ) = U ( i ) ( r , ω ) exp ( i ω t ) .
U ( i ) ( r , ω ) = | u | 2 1 a ( u , ω ) e i k u · r d 2 u .
W ( i ) ( r 1 , r 2 , ω ) = U ( i ) * ( r 1 , ω ) U ( i ) ( r 2 , ω ) ,
W ( i ) ( r 1 , r 2 , ω ) = | u | 2 1 | u | 2 1 A ( u , u , ω ) exp [ i k ( u · r 2 u · r 1 ) ] d 2 u d 2 u ,
A ( u , u , ω ) = a * ( u , ω ) a ( u , ω )
W ( 0 ) ( ρ 1 , ρ 2 , ω ) = [ S ( 0 ) ( ρ 1 , ω ) ] 1 / 2 [ S ( 0 ) ( ρ 2 , ω ) ] 1 / 2 μ ( 0 ) ( ρ 1 ρ 2 , ω ) ,
S ( 0 ) ( ρ , ω ) = A 0 2 exp ( ρ 2 / 2 σ S 2 ) ,
μ ( 0 ) ( ρ 1 ρ 2 , ω ) = exp [ ( ρ 2 ρ 1 ) 2 / 2 σ μ 2 ] .
A ( u , u , ω ) = ( k 2 π ) 4 + W ( 0 ) ( ρ 1 , ρ 2 , ω ) × exp [ i k ( u · ρ 2 u · ρ 1 ) ] d 2 ρ 1 d 2 ρ 2 .
U = ρ 2 ρ 1 ,
V = ( ρ 1 + ρ 2 ) / 2 ,
A ( u , u , ω ) = ( k 2 A 0 σ S σ eff 2 π ) 2 × exp { k 2 2 [ ( u u ) 2 σ S 2 + ( u + u ) 2 σ eff 2 4 ] } ,
1 σ eff 2 = 1 σ μ 2 + 1 4 σ S 2 .
1 σ μ 2 + 1 4 σ S 2 k 2 2 .
U ( s ) ( r u , ω ) f ( u , u 0 , ω ) e i k r r , ( k r , u   fixed ) ,
U ( s ) ( r u , ω ) e i k r r | u | 2 1 a ( u , ω ) f ( u , u , ω ) d 2 u .
J ( s ) ( u , ω ) = r 2 U ( s ) * ( r u , ω ) U ( s ) ( r u , ω ) , ( k r , with   u   fixed ) .
J ( s ) ( u , ω ) = A ( u , u , ω ) f * ( u , u , ω ) f ( u , u , ω ) d 2 u d 2 u .
J ( s ) ( u , ω ) = ( k 2 A 0 σ S σ eff 2 π ) 2 × exp { k 2 2 [ ( u u ) 2 σ S 2 + ( u + u ) 2 σ eff 2 4 ] } × f * ( u , u , ω ) f ( u , u , ω ) d 2 u d 2 u .
f ( u , u , ω ) = f ( u · u , ω ) = f ( θ , ω ) ,
f ( θ , ω ) = 1 k l = 0 ( 2 l + 1 ) exp [ i δ l ( ω ) ] sin δ l ( ω ) P l ( cos θ ) .
tan δ l ( ω ) = k ¯ j l ( k a ) j l ( k ¯ a ) k j l ( k ¯ a ) j l ( k a ) k ¯ j l ( k ¯ a ) n l ( k a ) k j l ( k ¯ a ) n l ( k a ) ,
k ¯ = n k ,
j l ( k a ) = d j l ( x ) d x x = k a ,
n l ( k a ) = d n l ( x ) d x x = k a .
J ( s ) ( u , ω ) = ( k A 0 σ S σ eff 2 π ) 2 l = 0 m = 0 b l * ( ω ) b m ( ω ) × exp { k 2 2 [ ( u u ) 2 σ S 2 + ( u + u ) 2 σ eff 2 4 ] } × P l ( u · u ) P m ( u · u ) d 2 u d 2 u ,
b m ( ω ) = ( 2 m + 1 ) exp [ i δ m ( ω ) ] sin δ m ( ω ) .
D e p f ( x , y ) g ( x , y ) d x d y 2 π g ( x 0 , y 0 ) p Det { H [ f ( x 0 , y 0 ) ] } e p f ( x 0 , y 0 ) , as     p ,
H [ f ( x 0 , y 0 ) ] = ( 2 f ( x , y ) x 2 2 f ( x , y ) x y 2 f ( x , y ) y x 2 f ( x , y ) y 2 ) x 0 , y 0 .
g ( u , u , u ) = ( k A 0 σ S σ eff 2 π ) 2 l = 0 m = 0 b l * ( ω ) b m ( ω ) P l ( u · u ) P m ( u · u ) exp { k 2 σ eff 2 8 ( u + u ) 2 } ,
f ( u , u ) = 1 2 ( u u ) 2 ,
p = ( k σ S ) 2 .
J ( s ) ( u , ω ) = A 0 2 σ μ 2 2 π l = 0 m = 0 b l * ( ω ) b m ( ω ) P l ( u · u ) P m ( u · u ) e k 2 σ μ 2 u 2 / 2 d 2 u .
J ( s ) ( u , ω ) = A 0 2 σ μ 2 2 π l = 0 m = 0 ( 2 l + 1 ) ( 2 m + 1 ) e i ( δ l δ m ) sin δ l sin δ m × | u | 2 1 P l ( u · u ) P m ( u · u ) e k 2 σ μ 2 u 2 / 2 d 2 u ,
P ( s ) ( σ μ , ω ) = ( 4 π ) J ( s ) ( u , ω ) d Ω .
P ( s ) ( σ μ , ω ) = 2 A 0 2 σ μ 2 l = 0 ( 2 l + 1 ) sin 2 δ l | u | 2 1 e k 2 σ μ 2 u 2 / 2 d 2 u ,
( 4 π ) P l ( u · u ) P m ( u · u ) d Ω = 4 π 2 l + 1 δ l m .
P ( s ) ( σ μ , ω ) = 4 π A 0 2 σ μ 2 l = 0 ( 2 l + 1 ) tan 2 δ l 1 + tan 2 δ l 0 π / 2 exp [ k 2 σ μ 2 sin 2 θ / 2 ] sin θ cos θ d θ = 4 π A 0 2 k 2 l = 0 ( 2 l + 1 ) tan 2 δ l 1 + tan 2 δ l [ 1 exp ( k 2 σ μ 2 2 ) ] .
P ( e ) ( ω ) = 4 π A 0 2 k f ( u 0 , u 0 ) ,
P ( s ) ( θ D , σ μ , ω ) = ( Ω D ) J ( s ) ( u , ω ) d Ω ( 4 π ) J ( s ) ( u , ω ) d Ω = 0 θ D J ( s ) ( θ , ω ) sin θ d θ 0 π J ( s ) ( θ , ω ) sin θ d θ .
J ( i ) ( u , ω ) = ( 2 π k ) 2 A ( u , u , ω ) cos 2 θ ,
= ( A 0 k σ S σ eff ) 2 cos 2 θ exp [ k 2 σ eff 2 sin 2 ( θ ) / 2 ] ,
P ( σ S , σ μ , ω ) = ( 2 π ) J ( i ) ( u , ω ) d Ω ,
P ( σ S , σ μ , ω ) = 2 π ( A 0 k σ S σ eff ) 2 × 0 π / 2 exp [ k 2 σ eff 2 sin 2 ( θ ) / 2 ] cos 2 θ sin θ d θ ,
= 2 π A 0 2 σ S 2 [ 1 exp ( k 2 σ eff 2 / 2 ) π / 2 erfi ( k σ eff / 2 ) k σ eff ] ,
erf ( x ) = 2 π 0 x exp ( t 2 ) d t .
P ( σ S , ω ) = 2 π A 0 2 σ S 2 , ( k σ eff 2 1 / 2 ) .

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