Abstract

We have developed a new technique based on light scattering experiments for tracing an alien particle with deterministic potential in a random collection of particles. We have shown that, via a sequence of measurements of light scattered to a far field of a scattering collection, it is possible to locate the center of the alien particle. The analysis of the stability of reconstruction is provided, and it is demonstrated via simulations that the results are stable for sufficiently large wavelength of the incident light and in cases when the size of the alien particle is comparable with the size of the typical particle in the collection.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).
  2. O. Korotkova and E. Wolf, “Scattering matrix theory for stochastic scalar fields,” Phys. Rev. E 75, 056609 (2007).
    [CrossRef]
  3. S. Sahin and O. Korotkova, “Scattering of scalar light fields from collections of particles,” Phys. Rev. A 78, 063815 (2008).
    [CrossRef]
  4. X. Du and D. Zhao, “Scattering of light by Gaussian-correlated quasi-homogeneous anisotropic media,” Opt. Lett. 35, 384–386(2010).
    [CrossRef] [PubMed]
  5. S. Sahin and O. Korotkova, “Effect of the pair-structure factor of a particulate medium on scalar wave scattering in the first Born approximation,” Opt. Lett. 34, 1762–1764 (2009).
    [CrossRef] [PubMed]
  6. T. Wang and D. Zhao, “Scattering theory of stochastic electromagnetic light waves,” Opt. Lett. 35, 2412–2414 (2010).
    [CrossRef] [PubMed]
  7. Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A 82, 033836 (2010).
    [CrossRef]
  8. M. Lahiri, E. Wolf, D. Fischer, and T. Shirai, “Determination of correlation functions of scattering potentials of stochastic media from scattering experiments,” Phys. Rev. Lett. 102, 123901 (2009).
    [CrossRef] [PubMed]
  9. T. Wang and D. Zhao, “Determination of pair-structure factor of scattering potential of a collection of particles,” Opt. Lett. 35, 318–320 (2010).
    [CrossRef] [PubMed]
  10. D. Zhao, O. Korotkova, and E. Wolf, “Application of correlation-induced spectral changes to inverse scattering,” Opt. Lett. 32, 3483–3485 (2007).
    [CrossRef] [PubMed]
  11. For example, the Fourier transform of the scattering potential of Gaussian type is a real function. See Eq. . For other scattering potentials with an exponential term of iK·r, such term will reduce to unity due to our requirement [Eq. ].
  12. T. D. Visser and E. Wolf, “Potential scattering with field discontinuities at the boundaries,” Phys. Rev. E 59, 2355–2360 (1999).
    [CrossRef]
  13. 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]

2010

Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A 82, 033836 (2010).
[CrossRef]

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. Wang and D. Zhao, “Determination of pair-structure factor of scattering potential of a collection of particles,” Opt. Lett. 35, 318–320 (2010).
[CrossRef] [PubMed]

X. Du and D. Zhao, “Scattering of light by Gaussian-correlated quasi-homogeneous anisotropic media,” Opt. Lett. 35, 384–386(2010).
[CrossRef] [PubMed]

T. Wang and D. Zhao, “Scattering theory of stochastic electromagnetic light waves,” Opt. Lett. 35, 2412–2414 (2010).
[CrossRef] [PubMed]

2009

M. Lahiri, E. Wolf, D. Fischer, and T. Shirai, “Determination of correlation functions of scattering potentials of stochastic media from scattering experiments,” Phys. Rev. Lett. 102, 123901 (2009).
[CrossRef] [PubMed]

S. Sahin and O. Korotkova, “Effect of the pair-structure factor of a particulate medium on scalar wave scattering in the first Born approximation,” Opt. Lett. 34, 1762–1764 (2009).
[CrossRef] [PubMed]

2008

S. Sahin and O. Korotkova, “Scattering of scalar light fields from collections of particles,” Phys. Rev. A 78, 063815 (2008).
[CrossRef]

2007

1999

T. D. Visser and E. Wolf, “Potential scattering with field discontinuities at the boundaries,” Phys. Rev. E 59, 2355–2360 (1999).
[CrossRef]

Du, X.

Fischer, D.

M. Lahiri, E. Wolf, D. Fischer, and T. Shirai, “Determination of correlation functions of scattering potentials of stochastic media from scattering experiments,” Phys. Rev. Lett. 102, 123901 (2009).
[CrossRef] [PubMed]

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]

Korotkova, O.

Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A 82, 033836 (2010).
[CrossRef]

S. Sahin and O. Korotkova, “Effect of the pair-structure factor of a particulate medium on scalar wave scattering in the first Born approximation,” Opt. Lett. 34, 1762–1764 (2009).
[CrossRef] [PubMed]

S. Sahin and O. Korotkova, “Scattering of scalar light fields from collections of particles,” Phys. Rev. A 78, 063815 (2008).
[CrossRef]

O. Korotkova and E. Wolf, “Scattering matrix theory for stochastic scalar fields,” Phys. Rev. E 75, 056609 (2007).
[CrossRef]

D. Zhao, O. Korotkova, and E. Wolf, “Application of correlation-induced spectral changes to inverse scattering,” Opt. Lett. 32, 3483–3485 (2007).
[CrossRef] [PubMed]

Lahiri, M.

M. Lahiri, E. Wolf, D. Fischer, and T. Shirai, “Determination of correlation functions of scattering potentials of stochastic media from scattering experiments,” Phys. Rev. Lett. 102, 123901 (2009).
[CrossRef] [PubMed]

Sahin, S.

Shirai, T.

M. Lahiri, E. Wolf, D. Fischer, and T. Shirai, “Determination of correlation functions of scattering potentials of stochastic media from scattering experiments,” Phys. Rev. Lett. 102, 123901 (2009).
[CrossRef] [PubMed]

Tong, Z.

Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A 82, 033836 (2010).
[CrossRef]

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 and E. Wolf, “Potential scattering with field discontinuities at the boundaries,” Phys. Rev. E 59, 2355–2360 (1999).
[CrossRef]

Wang, T.

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]

M. Lahiri, E. Wolf, D. Fischer, and T. Shirai, “Determination of correlation functions of scattering potentials of stochastic media from scattering experiments,” Phys. Rev. Lett. 102, 123901 (2009).
[CrossRef] [PubMed]

D. Zhao, O. Korotkova, and E. Wolf, “Application of correlation-induced spectral changes to inverse scattering,” Opt. Lett. 32, 3483–3485 (2007).
[CrossRef] [PubMed]

O. Korotkova and E. Wolf, “Scattering matrix theory for stochastic scalar fields,” Phys. Rev. E 75, 056609 (2007).
[CrossRef]

T. D. Visser and E. Wolf, “Potential scattering with field discontinuities at the boundaries,” Phys. Rev. E 59, 2355–2360 (1999).
[CrossRef]

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

Zhao, D.

Opt. Lett.

Phys. Rev. A

S. Sahin and O. Korotkova, “Scattering of scalar light fields from collections of particles,” Phys. Rev. A 78, 063815 (2008).
[CrossRef]

Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A 82, 033836 (2010).
[CrossRef]

Phys. Rev. E

O. Korotkova and E. Wolf, “Scattering matrix theory for stochastic scalar fields,” Phys. Rev. E 75, 056609 (2007).
[CrossRef]

T. D. Visser and E. Wolf, “Potential scattering with field discontinuities at the boundaries,” Phys. Rev. E 59, 2355–2360 (1999).
[CrossRef]

Phys. Rev. Lett.

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]

M. Lahiri, E. Wolf, D. Fischer, and T. Shirai, “Determination of correlation functions of scattering potentials of stochastic media from scattering experiments,” Phys. Rev. Lett. 102, 123901 (2009).
[CrossRef] [PubMed]

Other

For example, the Fourier transform of the scattering potential of Gaussian type is a real function. See Eq. . For other scattering potentials with an exponential term of iK·r, such term will reduce to unity due to our requirement [Eq. ].

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Normalized x component of the calculated position vector of the impurity with various e x determined from 18 randomly distributed normal particles.

Fig. 2
Fig. 2

Normalized x component of the calculated position vector of the impurity with various wavelengths determined from 18 randomly distributed normal particles.

Fig. 3
Fig. 3

Normalized x component of the calculated position vector of the impurity with various sizes of the impurity determined from 18 randomly distributed normal particles.

Fig. 4
Fig. 4

Normalized x component of the calculated position vector of the impurity with various wavelengths for different number of randomly distributed normal particles.

Equations (34)

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

U ( i ) ( r , s 0 ; ω ) = a ( ω ) exp ( i k s 0 · r ) ,
W ( i ) ( r 1 , r 2 , s 0 , ω ) = U ( i ) * ( r 1 , s 0 , ω ) U ( i ) ( r 2 , s 0 , ω ) ,
W ( i ) ( r 1 , r 2 , s 0 , ω ) = S ( i ) ( ω ) exp [ i k s 0 · ( r 2 r 1 ) ] ,
F ( r , ω ) = 1 4 π k 2 [ n 2 ( r , ω ) 1 ] ,
C F ( r 1 , r 2 , ω ) = F * ( r 1 , ω ) F ( r 2 , ω ) m ,
U ( s ) ( r s , ω ) = D F ( r , ω ) U ( i ) ( r , ω ) G ( | r s r | , ω ) d 3 r ,
G ( | r s r | , ω ) e i k r r e i k s · r .
W ( s ) ( r s 1 , r s 2 , s 0 , ω ) = U ( s ) * ( r s 1 , ω ) U ( s ) ( r s 2 , ω ) .
W ( s ) ( r s 1 , r s 2 , s 0 , ω ) = 1 r 2 S ( i ) ( ω ) C ˜ F [ k ( s 1 s 0 ) , k ( s 2 s 0 ) , ω ] ,
C ˜ F [ K 1 , K 2 , ω ] = D D C F ( r 1 , r 1 , ω ) exp [ i ( K 2 · r 2 K 1 · r 1 ) ] d 3 r 1 d 3 r 2
K 1 = k ( s 1 s 0 ) , K 2 = k ( s 2 s 0 ) .
f ˜ l ( K , ω ) = V f l ( r , ω ) e i K · r d 3 r ( l = 1 , 2 ) .
W ( s ) ( r s 1 , r s 2 , s 0 , ω ) = 1 r 2 S ( i ) ( ω ) F ( K 1 , ω ) M ( 2 ) ( K 1 , K 2 , ω ) F ( K 2 , ω ) ,
M ( 2 ) ( K 1 , K 2 , ω ) = ( Q 1 * ( K 1 , ω ) Q 1 ( K 2 , ω ) Q 1 * ( K 1 , ω ) Q 2 ( K 2 , ω ) Q 2 * ( K 1 , ω ) Q 1 ( K 2 , ω ) Q 2 * ( K 1 , ω ) Q 2 ( K 2 , ω ) )
C ˜ F ( K 1 , K 2 , ω ) = F ( K 1 , ω ) M ( 2 ) ( K 1 , K 2 , ω ) F ( K 2 , ω ) = r 2 W ( s ) ( r s 1 , r s 2 , s 0 , ω ) S ( i ) ( ω ) .
C ˜ F ( 0 ) ( K , K ) = f ˜ 1 * ( K ) f ˜ 1 ( K ) Q 1 * ( K ) Q 1 ( K ) ,
C ˜ F ( 1 ) ( K , K ) = f ˜ 1 * ( K ) f ˜ 1 ( K ) Q 1 * ( K ) Q 1 ( K ) + f ˜ 1 * ( K ) f ˜ 2 ( K ) Q 1 * ( K ) + f ˜ 2 * ( K ) f ˜ 1 ( K ) Q 1 ( K ) + f ˜ 2 * ( K ) f ˜ 2 ( K ) ,
Re ( Q 1 ( K ) ) = C ˜ F ( 1 ) ( K , K ) C ˜ F ( 0 ) ( K , K ) 2 f ˜ 1 ( K ) f ˜ 2 ( K ) f ˜ 2 ( K ) 2 f ˜ 1 ( K ) .
C ˜ F ( 2 ) ( K , K ) = f ˜ 1 * ( K ) f ˜ 1 ( K ) Q 1 * ( K ) Q 1 ( K ) + f ˜ 1 * ( K ) f ˜ 2 ( K ) Q 1 * ( K ) e i K · r 2 + f ˜ 2 * ( K ) f ˜ 1 ( K ) Q 1 ( K ) e i K · r 2 + f ˜ 2 * ( K ) f ˜ 2 ( K ) ,
Im ( Q 1 ( K ) ) = Re ( Q 1 ( K ) ) cot ( K · r 2 ) C ˜ F ( 2 ) ( K , K ) C ˜ F ( 0 ) ( K , K ) 2 f ˜ 1 ( K ) f ˜ 2 ( K ) sin ( K · r 2 ) + f ˜ 2 ( K ) 2 f ˜ 1 ( K ) sin ( K · r 2 ) .
Q 1 ( K ) = Re ( Q 1 ( K ) ) + i Im ( Q 1 ( K ) ) .
C ˜ F ( i m ) ( K , K ) = f ˜ 1 * ( K ) f ˜ 1 ( K ) Q 1 * ( K ) Q 1 ( K ) + f ˜ 1 * ( K ) f ˜ 0 ( K ) Q 1 * ( K ) e i K · r 0 + f ˜ 0 * ( K ) f ˜ 1 ( K ) Q 1 ( K ) e i K · r 0 + f ˜ 0 * ( K ) f ˜ 0 ( K ) ,
Re ( Q 1 ( K ) ) cos ( K · r 0 ) Im ( Q 1 ( K ) ) sin ( K · r 0 ) = C ˜ F ( i m ) ( K , K ) C ˜ F ( 0 ) ( K , K ) 2 f ˜ 1 ( K ) f ˜ 0 ( K ) f ˜ 0 ( K ) 2 f ˜ 1 ( K ) C ( K ) .
| e x | 1 , | 1 e z | 1 , | 1 e z | | e x | .
cos ( K · r 0 ) 1 1 2 ( K · r ) 2 , sin ( K · r 0 ) K · r ,
x = Im ( Q 1 ( K ) ) + Im 2 ( Q 1 ( K ) ) 2 Re ( Q 1 ( K ) ) [ C ( K ) Re ( Q 1 ( K ) ) ] k e x Re ( Q 1 ( K ) ) ,
x ¯ = Im ( Q 1 ( K ) ) + Im 2 ( Q 1 ( K ) ) 2 Re ( Q 1 ( K ) ) [ C ( N ) ( K ) Re ( Q 1 ( K ) ) ] k e x Re ( Q 1 ( K ) ) ,
C ( N ) ( K ) = C ˜ F ( i m ) ( K , K ) C ˜ F ( o r ) ( K , K ) 2 N f ˜ 1 ( K ) f ˜ 0 ( K ) N f ˜ 0 ( K ) 2 f ˜ 1 ( K ) .
| 1 24 ( K · r 0 ) 4 · Re ( Q 1 ( K ) ) | | K · r 0 · Im ( Q 1 ( K ) ) | ,
| Im ( Q 1 ( K ) ) · 1 6 ( K · r 0 ) 3 | | 1 2 ( K · r 0 ) 2 · Re ( Q 1 ( K ) ) | ,
| Im ( Q 1 ( K ) ) Re ( Q 1 ( K ) ) | 1 24 | K · r 0 | 3 ,
| Im ( Q 1 ( K ) ) Re ( Q 1 ( K ) ) | 3 | K · r 0 | 1 .
f i ( r , ω ) = B exp [ r 2 2 σ i 2 ] ( i = 0 , 1 , 2 ) ,
f ˜ i ( K , ω ) = B ( 2 π ) ( 3 / 2 ) σ i 3 exp [ K 2 σ i 2 / 2 ] ( i = 0 , 1 , 2 ) .

Metrics