Abstract

This paper uses the Cramér–Rao bound (CRB) to investigate the role of multiple scattering in the framework of transverse electric (TE) electromagnetic inverse scattering. The surface mode in the TE scattering case enables a strong multiple scattering effect even if the scatterers are widely separated. We show that multiple scattering does not always improve the accuracy of the estimation, compared with that expected with a fictitious single scattering model. The reason for the aforementioned conclusion is discussed. Two scenarios that favor the multiple scattering effect are discussed, and the differences between these scenarios and the CRB analysis are highlighted.

© 2010 Optical Society of America

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References

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  1. J. de Rosny and C. Prada, “Comment on 'Multiple scattering: the key to unravel the subwavelength world from the far-field pattern of a scattered wave,'” Phys. Rev. E 75, 048601 (2007).
    [CrossRef]
  2. A. Sentenac, C. A. Guerin, P. C. Chaumet, F. Drsek, H. Giovannini, N. Bertaux, and M. Holschneider, “Influence of multiple scattering on the resolution of an imaging system: a Cramér-Rao analysis,” Opt. Express 15, 1340-1347 (2007).
    [CrossRef] [PubMed]
  3. F. Simonetti, M. Fleming, and E. A. Marengo, “Illustration of the role of multiple scattering in subwavelength imaging from far-field measurements,” J. Opt. Soc. Am. A 25, 292-303 (2008).
    [CrossRef]
  4. K. Agarwal and X. Chen, “Applicability of MUSIC-type imaging in two-dimensional electromagnetic inverse problems,” IEEE Trans. Antennas Propag. 56, 3217-3223 (2008).
    [CrossRef]
  5. A. Kirsch, “The MUSIC-algorithm and the factorization method in inverse scattering theory for inhomogeneous media,” Inverse Probl. 18, 1025-1040 (2002).
    [CrossRef]
  6. C. F. Bohren and D. R. Huffman, Absorption and Scatttering of Light by Small Particles (Wiley, 1983).
  7. I. Tolstoy and A. Tolstoy, “Superresonant systems of scatterers. II,” J. Acoust. Soc. Am. 83, 2086-2096 (1988).
    [CrossRef]
  8. L. Borcea, G. Papanicolaou, C. Tsogka, and J. Berryman, “Imaging and time reversal in random media,” Inverse Probl. 18, 1247-1279 (2002).
    [CrossRef]
  9. G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-field time reversal,” Science 315, 1120-1122 (2007).
    [CrossRef] [PubMed]
  10. R. Carminati, R. Pierrat, J. de Rosny, and M. Fink, “Theory of the time reversal cavity for electromagnetic fields,” Opt. Lett. 32, 3107-3109 (2007).
    [CrossRef] [PubMed]
  11. F. C. Chen and W. C. Chew, “Experimental verification of super resolution in nonlinear inverse scattering,” Appl. Phys. Lett. 72, 3080-3082 (1998).
    [CrossRef]

2008 (2)

F. Simonetti, M. Fleming, and E. A. Marengo, “Illustration of the role of multiple scattering in subwavelength imaging from far-field measurements,” J. Opt. Soc. Am. A 25, 292-303 (2008).
[CrossRef]

K. Agarwal and X. Chen, “Applicability of MUSIC-type imaging in two-dimensional electromagnetic inverse problems,” IEEE Trans. Antennas Propag. 56, 3217-3223 (2008).
[CrossRef]

2007 (4)

J. de Rosny and C. Prada, “Comment on 'Multiple scattering: the key to unravel the subwavelength world from the far-field pattern of a scattered wave,'” Phys. Rev. E 75, 048601 (2007).
[CrossRef]

A. Sentenac, C. A. Guerin, P. C. Chaumet, F. Drsek, H. Giovannini, N. Bertaux, and M. Holschneider, “Influence of multiple scattering on the resolution of an imaging system: a Cramér-Rao analysis,” Opt. Express 15, 1340-1347 (2007).
[CrossRef] [PubMed]

G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-field time reversal,” Science 315, 1120-1122 (2007).
[CrossRef] [PubMed]

R. Carminati, R. Pierrat, J. de Rosny, and M. Fink, “Theory of the time reversal cavity for electromagnetic fields,” Opt. Lett. 32, 3107-3109 (2007).
[CrossRef] [PubMed]

2002 (2)

A. Kirsch, “The MUSIC-algorithm and the factorization method in inverse scattering theory for inhomogeneous media,” Inverse Probl. 18, 1025-1040 (2002).
[CrossRef]

L. Borcea, G. Papanicolaou, C. Tsogka, and J. Berryman, “Imaging and time reversal in random media,” Inverse Probl. 18, 1247-1279 (2002).
[CrossRef]

1998 (1)

F. C. Chen and W. C. Chew, “Experimental verification of super resolution in nonlinear inverse scattering,” Appl. Phys. Lett. 72, 3080-3082 (1998).
[CrossRef]

1988 (1)

I. Tolstoy and A. Tolstoy, “Superresonant systems of scatterers. II,” J. Acoust. Soc. Am. 83, 2086-2096 (1988).
[CrossRef]

Agarwal, K.

K. Agarwal and X. Chen, “Applicability of MUSIC-type imaging in two-dimensional electromagnetic inverse problems,” IEEE Trans. Antennas Propag. 56, 3217-3223 (2008).
[CrossRef]

Berryman, J.

L. Borcea, G. Papanicolaou, C. Tsogka, and J. Berryman, “Imaging and time reversal in random media,” Inverse Probl. 18, 1247-1279 (2002).
[CrossRef]

Bertaux, N.

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scatttering of Light by Small Particles (Wiley, 1983).

Borcea, L.

L. Borcea, G. Papanicolaou, C. Tsogka, and J. Berryman, “Imaging and time reversal in random media,” Inverse Probl. 18, 1247-1279 (2002).
[CrossRef]

Carminati, R.

Chaumet, P. C.

Chen, F. C.

F. C. Chen and W. C. Chew, “Experimental verification of super resolution in nonlinear inverse scattering,” Appl. Phys. Lett. 72, 3080-3082 (1998).
[CrossRef]

Chen, X.

K. Agarwal and X. Chen, “Applicability of MUSIC-type imaging in two-dimensional electromagnetic inverse problems,” IEEE Trans. Antennas Propag. 56, 3217-3223 (2008).
[CrossRef]

Chew, W. C.

F. C. Chen and W. C. Chew, “Experimental verification of super resolution in nonlinear inverse scattering,” Appl. Phys. Lett. 72, 3080-3082 (1998).
[CrossRef]

de Rosny, J.

R. Carminati, R. Pierrat, J. de Rosny, and M. Fink, “Theory of the time reversal cavity for electromagnetic fields,” Opt. Lett. 32, 3107-3109 (2007).
[CrossRef] [PubMed]

G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-field time reversal,” Science 315, 1120-1122 (2007).
[CrossRef] [PubMed]

J. de Rosny and C. Prada, “Comment on 'Multiple scattering: the key to unravel the subwavelength world from the far-field pattern of a scattered wave,'” Phys. Rev. E 75, 048601 (2007).
[CrossRef]

Drsek, F.

Fink, M.

R. Carminati, R. Pierrat, J. de Rosny, and M. Fink, “Theory of the time reversal cavity for electromagnetic fields,” Opt. Lett. 32, 3107-3109 (2007).
[CrossRef] [PubMed]

G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-field time reversal,” Science 315, 1120-1122 (2007).
[CrossRef] [PubMed]

Fleming, M.

Giovannini, H.

Guerin, C. A.

Holschneider, M.

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scatttering of Light by Small Particles (Wiley, 1983).

Kirsch, A.

A. Kirsch, “The MUSIC-algorithm and the factorization method in inverse scattering theory for inhomogeneous media,” Inverse Probl. 18, 1025-1040 (2002).
[CrossRef]

Lerosey, G.

G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-field time reversal,” Science 315, 1120-1122 (2007).
[CrossRef] [PubMed]

Marengo, E. A.

Papanicolaou, G.

L. Borcea, G. Papanicolaou, C. Tsogka, and J. Berryman, “Imaging and time reversal in random media,” Inverse Probl. 18, 1247-1279 (2002).
[CrossRef]

Pierrat, R.

Prada, C.

J. de Rosny and C. Prada, “Comment on 'Multiple scattering: the key to unravel the subwavelength world from the far-field pattern of a scattered wave,'” Phys. Rev. E 75, 048601 (2007).
[CrossRef]

Sentenac, A.

Simonetti, F.

Tolstoy, A.

I. Tolstoy and A. Tolstoy, “Superresonant systems of scatterers. II,” J. Acoust. Soc. Am. 83, 2086-2096 (1988).
[CrossRef]

Tolstoy, I.

I. Tolstoy and A. Tolstoy, “Superresonant systems of scatterers. II,” J. Acoust. Soc. Am. 83, 2086-2096 (1988).
[CrossRef]

Tourin, A.

G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-field time reversal,” Science 315, 1120-1122 (2007).
[CrossRef] [PubMed]

Tsogka, C.

L. Borcea, G. Papanicolaou, C. Tsogka, and J. Berryman, “Imaging and time reversal in random media,” Inverse Probl. 18, 1247-1279 (2002).
[CrossRef]

Appl. Phys. Lett. (1)

F. C. Chen and W. C. Chew, “Experimental verification of super resolution in nonlinear inverse scattering,” Appl. Phys. Lett. 72, 3080-3082 (1998).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

K. Agarwal and X. Chen, “Applicability of MUSIC-type imaging in two-dimensional electromagnetic inverse problems,” IEEE Trans. Antennas Propag. 56, 3217-3223 (2008).
[CrossRef]

Inverse Probl. (2)

A. Kirsch, “The MUSIC-algorithm and the factorization method in inverse scattering theory for inhomogeneous media,” Inverse Probl. 18, 1025-1040 (2002).
[CrossRef]

L. Borcea, G. Papanicolaou, C. Tsogka, and J. Berryman, “Imaging and time reversal in random media,” Inverse Probl. 18, 1247-1279 (2002).
[CrossRef]

J. Acoust. Soc. Am. (1)

I. Tolstoy and A. Tolstoy, “Superresonant systems of scatterers. II,” J. Acoust. Soc. Am. 83, 2086-2096 (1988).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. E (1)

J. de Rosny and C. Prada, “Comment on 'Multiple scattering: the key to unravel the subwavelength world from the far-field pattern of a scattered wave,'” Phys. Rev. E 75, 048601 (2007).
[CrossRef]

Science (1)

G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-field time reversal,” Science 315, 1120-1122 (2007).
[CrossRef] [PubMed]

Other (1)

C. F. Bohren and D. R. Huffman, Absorption and Scatttering of Light by Small Particles (Wiley, 1983).

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Figures (4)

Fig. 1
Fig. 1

Comparison of CRBs of the MS model and SS model. Only x 1 and x 2 are the parameters to be estimated. There is a single incidence with k ̂ i = x ̂ , and 20 receivers are distributed on a circle. (a) CRB ( x 1 ) , (b) CRB ( x 2 ) .

Fig. 2
Fig. 2

Comparison of CRBs of the MS model and SS model. Only x 1 and x 2 are the parameters to be estimated. 20 transmitters and 20 receivers are uniformly distributed on a circle. (a) CRB ( x 1 ) , (b) CRB ( x 2 ) .

Fig. 3
Fig. 3

Same as Fig. 2 except there are eight unknowns.

Fig. 4
Fig. 4

Comparison of CRBs of the MS model and SS model with eight unknowns. There is a single incidence with k ̂ i = x ̂ , and 20 receivers are uniformly distributed in 0.3 π ϕ 0.7 π . (a) CRB ( x 1 ) , (b) CRB ( x 2 ) .

Equations (8)

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H zt ( p ) ( r 1 ) = H zi ( p ) ( r 1 ) + z ̂ { g ( r 1 , r 2 ) × [ ξ 2 E tt ( p ) ( r 2 ) ] } ,
E tt ( p ) ( r 1 ) = E ti ( p ) ( r 1 ) + i k η 0 G ̿ ( r 1 , r 2 ) [ ξ 2 E tt ( p ) ( r 2 ) ] ,
H zt ( p ) ( r 2 ) = H zi ( p ) ( r 2 ) + z ̂ { g ( r 2 , r 1 ) × [ ξ 1 E tt ( p ) ( r 1 ) ] } ,
E tt ( p ) ( r 2 ) = E ti ( p ) ( r 2 ) + i k η 0 G ̿ ( r 2 , r 1 ) [ ξ 1 E tt ( p ) ( r 1 ) ] ,
H z ( p ) ( r q s ) = j = 1 2 z ̂ { g ( r q s , r j ) × [ ξ j E tt ( p ) ( r j ) ] } .
F i , j ( θ ) = 2 σ 2 R [ ( K ¯ ( θ ) θ i ) * K ¯ ( θ ) θ j ] , i , j = 1 , 2 , , 8 ,
H z ( ϕ j ) = α cos ϕ j e i k x 1 cos ϕ j ξ 1 H 1 + α cos ϕ j e i k x 2 cos ϕ j ξ 2 e i k x 2 ,
H z ( ϕ j ) x 1 = α cos ϕ j ξ 1 k e i k x 1 cos ϕ j [ i ( cos ϕ j + 1 ) H 1 + ( e i k x 1 H 1 ) ( i G y y G y y ) ] ,

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