Abstract

We revisit the notion of resolution of an imaging system in the light of a probabilistic concept, the Cramér-Rao bound (CRB). We show that the CRB provides a simple quantitative estimation of the accuracy one can expect in measuring an unknown parameter from a scattering experiment. We then investigate the influence of multiple scattering on the CRB for the estimation of the interdistance between two objects in a typical two-sphere scattering experiments. We show that, contrarily to a common belief, the occurence of strong multiple scattering does not automatically lead to a resolution enhancement.

© 2007 Optical Society of America

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References

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  1. F. Simonetti, “Multiple scattering: The key to unravel the subwavelength world from the far-field pattern of a scattered wave,” Phys. Rev. E 73, (2006).
    [CrossRef]
  2. F.C. Chen and W.C. Chew, “Experimental verification of super resolution in nonlinear inverse scattering,” Appl. Phys. Lett. 72,3080–3082 (1998).
    [CrossRef]
  3. K Belkebir, A Sentenac, and PC Chaumet, “Influence of multiple scattering on three-dimensional imaging with optical diffraction tomography,” J. Opt. Soc. Am. A 23,586–595 (2006).
    [CrossRef]
  4. P. Blomgren and G. Papanicolaou, “Super-resolution in time-reversal acoustics,” J. Acoust. Soc. Am. 111,230–248 (2002).
    [CrossRef] [PubMed]
  5. C. Prada and J.L. Thomas, “Experimental subwavelength localization of scatterers by decomposition of time-reversal operator,” J. Acoust. Soc. Am. 114,235–243 (2003).
    [CrossRef] [PubMed]
  6. M. Shahram and P. Milanfar, “Imaging below the diffraction limit: A statistical analysis,” IEEE Trans. Image Proc. 13,677–689 (2004).
    [CrossRef]
  7. S. Van Aert, D. Van Dyck, and A.J. den Dekker, “Resolution of coherent and incoherent imaging systems reconsidered - classical criteria and a statistical alternative,” Opt. Express 14,3830–3839 (2006).
    [CrossRef] [PubMed]
  8. Refregier P., “Noise theory and application to physics from fluctuation to information,” chapter statistical estimation,167–205, springer 2004.
  9. Y.L. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34,4573–4588 (1995).
    [CrossRef] [PubMed]
  10. P.C. Chaumet and M. Nieto-Vesperinas, “Optical binding of particles with or without the presence of a flat dielectric surface,” Phys. Rev. B 64,035422–035429 (2001).
    [CrossRef]

2006 (3)

2004 (1)

M. Shahram and P. Milanfar, “Imaging below the diffraction limit: A statistical analysis,” IEEE Trans. Image Proc. 13,677–689 (2004).
[CrossRef]

2003 (1)

C. Prada and J.L. Thomas, “Experimental subwavelength localization of scatterers by decomposition of time-reversal operator,” J. Acoust. Soc. Am. 114,235–243 (2003).
[CrossRef] [PubMed]

2002 (1)

P. Blomgren and G. Papanicolaou, “Super-resolution in time-reversal acoustics,” J. Acoust. Soc. Am. 111,230–248 (2002).
[CrossRef] [PubMed]

2001 (1)

P.C. Chaumet and M. Nieto-Vesperinas, “Optical binding of particles with or without the presence of a flat dielectric surface,” Phys. Rev. B 64,035422–035429 (2001).
[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]

1995 (1)

Aert, S. Van

Belkebir, K

Blomgren, P.

P. Blomgren and G. Papanicolaou, “Super-resolution in time-reversal acoustics,” J. Acoust. Soc. Am. 111,230–248 (2002).
[CrossRef] [PubMed]

Chaumet, P.C.

P.C. Chaumet and M. Nieto-Vesperinas, “Optical binding of particles with or without the presence of a flat dielectric surface,” Phys. Rev. B 64,035422–035429 (2001).
[CrossRef]

Chaumet, PC

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]

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]

Dekker, A.J. den

Dyck, D. Van

Milanfar, P.

M. Shahram and P. Milanfar, “Imaging below the diffraction limit: A statistical analysis,” IEEE Trans. Image Proc. 13,677–689 (2004).
[CrossRef]

Nieto-Vesperinas, M.

P.C. Chaumet and M. Nieto-Vesperinas, “Optical binding of particles with or without the presence of a flat dielectric surface,” Phys. Rev. B 64,035422–035429 (2001).
[CrossRef]

P., Refregier

Refregier P., “Noise theory and application to physics from fluctuation to information,” chapter statistical estimation,167–205, springer 2004.

Papanicolaou, G.

P. Blomgren and G. Papanicolaou, “Super-resolution in time-reversal acoustics,” J. Acoust. Soc. Am. 111,230–248 (2002).
[CrossRef] [PubMed]

Prada, C.

C. Prada and J.L. Thomas, “Experimental subwavelength localization of scatterers by decomposition of time-reversal operator,” J. Acoust. Soc. Am. 114,235–243 (2003).
[CrossRef] [PubMed]

Sentenac, A

Shahram, M.

M. Shahram and P. Milanfar, “Imaging below the diffraction limit: A statistical analysis,” IEEE Trans. Image Proc. 13,677–689 (2004).
[CrossRef]

Simonetti, F.

F. Simonetti, “Multiple scattering: The key to unravel the subwavelength world from the far-field pattern of a scattered wave,” Phys. Rev. E 73, (2006).
[CrossRef]

Thomas, J.L.

C. Prada and J.L. Thomas, “Experimental subwavelength localization of scatterers by decomposition of time-reversal operator,” J. Acoust. Soc. Am. 114,235–243 (2003).
[CrossRef] [PubMed]

Xu, Y.L.

Appl. Opt. (1)

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. Image Proc. (1)

M. Shahram and P. Milanfar, “Imaging below the diffraction limit: A statistical analysis,” IEEE Trans. Image Proc. 13,677–689 (2004).
[CrossRef]

J. Acoust. Soc. Am. (2)

P. Blomgren and G. Papanicolaou, “Super-resolution in time-reversal acoustics,” J. Acoust. Soc. Am. 111,230–248 (2002).
[CrossRef] [PubMed]

C. Prada and J.L. Thomas, “Experimental subwavelength localization of scatterers by decomposition of time-reversal operator,” J. Acoust. Soc. Am. 114,235–243 (2003).
[CrossRef] [PubMed]

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

Opt. Express (1)

Phys. Rev. B (1)

P.C. Chaumet and M. Nieto-Vesperinas, “Optical binding of particles with or without the presence of a flat dielectric surface,” Phys. Rev. B 64,035422–035429 (2001).
[CrossRef]

Phys. Rev. E (1)

F. Simonetti, “Multiple scattering: The key to unravel the subwavelength world from the far-field pattern of a scattered wave,” Phys. Rev. E 73, (2006).
[CrossRef]

Other (1)

Refregier P., “Noise theory and application to physics from fluctuation to information,” chapter statistical estimation,167–205, springer 2004.

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Figures (5)

Fig. 1.
Fig. 1.

The set-up of the scattering experiment. In the first configuration, the spheres are aligned in the direction of the incident wave vector K 0. The scattered waves (K) are recorded in a 30 degree aperture cone in the scattering plane (ẑ,x̂) around the forward (A) or backward (B) direction. The incident polarization E 0 is perpendicular to the scattering plane. In the second configuration, the sphere alignment is perpendicular to K 0 and only the backward direction is investigated. The polarization E 0 is perpendicular (C1) or parallel (C2) to the scattering plane.

Fig. 2.
Fig. 2.

Relative CRB predicted by a single scattering analysis in different configurations for small spheres (d=0.06λ) and two different indices n=1.1 and n=3.1. The proportionality constants in eq. (10) and (11) have been taken to 1. Changing the index results in a vertical translation of the CRB in the case of additive noise (top), but has no influence in the case of multiplicative noise (bottom).

Fig. 4.
Fig. 4.

Same as Fig. 3 in configuration A (bottom) and B (top). The peaks are due to the zeros of intensity, that make the inverse CRB explode (see eq. 7)

Fig. 3.
Fig. 3.

Evolution of the relative CRB as a function of the normalized sphere interdistance α/λ, in configuration C1 (top) and C2 (bottom). The spheres have diameter 0.06λ and the noise is multiplicative. The curve labelled ‘ni’ (for ‘no interaction’) is the result of ignoring the coupling between the spheres, for the three optical indices.

Fig. 5.
Fig. 5.

Same as Fig. 4 for spheres of diameter 0.3λ. The coupling between the spheres is visible on a wider range as for small spheres. In (a), the peaks occur, again, at the distances that lead to destructive interference between the spheres.

Equations (12)

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E [ ( α ̂ α ) 2 ] ( E [ α L ( α ) ] 2 ) 1 ,
I j m = I j t + N j , j = 1 , . . , N ,
N j ~ N ( 0 , I 0 2 ) , E [ N i N j ] = I 0 2 δ ij ,
CRB = I 0 2 ( j = 1 N [ α I j t ] 2 ) 1
I j m = I j t N j , j = 1 , . . , N .
N j ~ Γ ( μ , L ) , E [ N i N j ] = μ 2 L δ ij + μ 2 .
CRB = μ 2 L ( j = 1 N [ α I j t I j t ] 2 ) 1
Resolution ( α ) = CRB α 2
I ( K ̂ ) = 2 I s ( K ̂ ) ( 1 + cos [ α Φ ( K ̂ ) ] )
CRB 1 const × Φ min Φ max Φ 2 sin 2 ( α Φ ) 1 + cos ( α Φ ) 2
CRB 1 const × I s Φ min Φ max Φ 2 sin 2 ( α Φ )
I ( K ̂ ) ~ χ ̂ ( K K 0 ) 2 ,

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