Abstract

Recently it has been proposed that the classical diffraction limit could be overcome by taking into account multiple scattering effects to describe the interaction of a probing wave and the object to be imaged [Phys. Rev. E 73, 036619 (2006) ]. Here this idea is illustrated by considering two point scatterers spaced much less than a wavelength apart. It is observed that while under the Born approximation the scattered field pattern is similar to that of a monopole source centered between the scatterers, multiple scattering leads to a more complicated pattern. This additional complexity carries information about the subwavelength structure and can lead to superresolution in the presence of large noise levels. Moreover, it is pointed out that the additional information due to multiple scattering is interpreted as a form of coherent noise by inversion algorithms based on the Born approximation.

© 2008 Optical Society of America

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References

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  1. E. Wolf, "Three-dimensional structure determination of semi-transparent objects from holographic data," Opt. Commun. 1, 153-156 (1969).
    [CrossRef]
  2. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).
  3. F. Simonetti, "Multiple scattering: The key to unravel the subwavelength world from the far-field pattern of a scattered wave," Phys. Rev. E 73, 036619-1 (2006).
    [CrossRef]
  4. T. J. Cui, W. C. Chew, X. X. Yin, and W. Hong, "Study of resolution and super resolution in electromagnetic imaging for half-space problems," IEEE Trans. Antennas Propag. 52, 1398-1411 (2004).
    [CrossRef]
  5. K. Belkebir, P. C. Chaumet, and A. Sentenac, "Influence of multiple scattering on three-dimensional imaging with optical diffraction tomography," J. Opt. Soc. Am. A 23, 586-595 (2006).
    [CrossRef]
  6. F. C. Chen and W. C. Chew, "Experimental verifiction of super resolution in nonlinear inverse scattering," Appl. Phys. Lett. 72, 3080-3082 (1998).
    [CrossRef]
  7. F. Simonetti, "Localization of point-like scatterers in solids with subwavelength resolution," Appl. Phys. Lett. 89, 094105 (2006).
    [CrossRef]
  8. F. Simonetti, L. Huang, N. Duric, and O. Rama, "Imaging beyond the Born approximation: An experimental investigation with an ultrasonic ring array," Phys. Rev. E 76, 036601 (2007).
    [CrossRef]
  9. E. A. Marengo and F. K. Gruber, "Subspace-based localization and inverse scattering of multiply scattering point targets," EURASIP J. Adv. Sign. Processing 2007, ID 17324 16 pp. (Hindawi Pub. Corp., 2007).
  10. 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]
  11. F. Simonetti, "Reply to 'comment on "Multiple scattering: the key to unravel the subwavelength world from the far-field pattern of a scattered wave"'," Phys. Rev. E 75, 048602 (2007).
    [CrossRef]
  12. L. L. Foldy, "The multiple scattering of waves. i. general theory of isotropic scattering by randomly distributed scatterers," Phys. Rev. 67, 107-119 (1945).
    [CrossRef]
  13. D. Colton and R. Kress, "Using fundamental solutions in inverse scattering," Inverse Probl. 22, R49-66 (2006).
    [CrossRef]
  14. A. Kirsch, "Characterization of the shape of a scattering obstacle using the spectral data of the far field operator," Inverse Probl. 14, 1489-1512 (1998).
    [CrossRef]
  15. S. Hou, K. Solna, and H. Zhao, "A direct imaging algorithm for extended targets," Inverse Probl. 22, 1151-1178 (2006).
    [CrossRef]
  16. E. A. Marengo, F. K. Gruber, and F. Simonetti, "Time-reversal music imaging of extended targets," IEEE Trans. Image Process. 16, 1967-1984 (2007).
    [CrossRef] [PubMed]
  17. R. Potthast, Point Sources and Multipoles in Inverse Scattering Theory (Chapman & Hall / CRC, 2001).
    [CrossRef]
  18. J. Hadamard, Lectures on Cauchy's Problems in Linear Partial Differential Equations (Yale U. Press, 1923).
  19. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
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    [CrossRef]
  22. G. Toraldo Di Francia, "Degrees of freedom of an image," J. Opt. Soc. Am. 59, 799-804 (1969).
    [CrossRef] [PubMed]
  23. T. Habashy and E. Wolf, "Reconstruction of scattering potentials from incomplete data," J. Mod. Opt. 41, 1679-1685 (1994).
    [CrossRef]
  24. M. Bertero and P. Boccacci, "Super-resolution in computational imaging," Micron 34, 265-273 (2003).
    [CrossRef] [PubMed]
  25. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).
  26. V. Twersky, "Multiple scattering of waves and optical phenomena," J. Opt. Soc. Am. 52, 145-171 (1961).
    [CrossRef]
  27. M. Cheney and R. Bonneau, "Imaging that exploits multipath scattering from point scatterers," Inverse Probl. 20, 1691-1711 (2004).
    [CrossRef]
  28. I. Tolstoy, "Supreresonant systems of scatterers. i," J. Acoust. Soc. Am. 80, 282-294 (1986).
    [CrossRef]
  29. I. Tolstoy and A. Tolstoy, "Supreresonant systems of scatterers. ii," J. Acoust. Soc. Am. 83, 2086-2096 (1988).
    [CrossRef]
  30. E. J. Heller, "Quantum proximity resonance," Phys. Rev. Lett. 77, 4122-4125 (1996).
    [CrossRef] [PubMed]
  31. J. S. Hersch and E. J. Heller, "Observation of proximity resonances in a parallel-plate waveguide," Phys. Rev. Lett. 81, 3059-3062 (1998).
    [CrossRef]
  32. A. J. Devaney, "Super-resolution processing of multi-static data using time-reversal and music," unpublished; available at www.ece.neu.edu/faculty/devaney/ajd/preprints.htm (2000).
  33. H. Lev-Ari and A. J. Devaney, "The time-reversal technique re-interpreted: subspace-based signal processing for multi-static target location," in Proceedings of the 2000 IEEE Sensor Array and Multichannel Signal Processing Workshop (IEEE, 2000), pp. 509-513.
  34. F. K. Gruber, E. A. Marengo, and A. J. Devaney, "Time-reversal imaging with multiple signal classification considering multiple scattering between the targets," J. Acoust. Soc. Am. 115, 3042-3047 (2004).
    [CrossRef]
  35. G. Shi and A. Nehorai, "Maximum likelihood estimation of point scatterers for computational time-reversal imaging," Commun. Inf. Syst. 5, 227-256 (2005).
  36. M. Rusek, J. Mostowski, and A. Orlowski, "Random Green matrices: From proximity resonance to Anderson localization," Phys. Rev. A 61, 022704 (2000).
    [CrossRef]
  37. R. K. Snieder and J. A. Scales, "Time-reversed imaging as a diagnostic of wave and particle chaos," Phys. Rev. E 58, 5668-5675 (1998).
    [CrossRef]

2007 (4)

F. Simonetti, L. Huang, N. Duric, and O. Rama, "Imaging beyond the Born approximation: An experimental investigation with an ultrasonic ring array," Phys. Rev. E 76, 036601 (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]

F. Simonetti, "Reply to 'comment on "Multiple scattering: the key to unravel the subwavelength world from the far-field pattern of a scattered wave"'," Phys. Rev. E 75, 048602 (2007).
[CrossRef]

E. A. Marengo, F. K. Gruber, and F. Simonetti, "Time-reversal music imaging of extended targets," IEEE Trans. Image Process. 16, 1967-1984 (2007).
[CrossRef] [PubMed]

2006 (5)

S. Hou, K. Solna, and H. Zhao, "A direct imaging algorithm for extended targets," Inverse Probl. 22, 1151-1178 (2006).
[CrossRef]

F. Simonetti, "Localization of point-like scatterers in solids with subwavelength resolution," Appl. Phys. Lett. 89, 094105 (2006).
[CrossRef]

D. Colton and R. Kress, "Using fundamental solutions in inverse scattering," Inverse Probl. 22, R49-66 (2006).
[CrossRef]

F. Simonetti, "Multiple scattering: The key to unravel the subwavelength world from the far-field pattern of a scattered wave," Phys. Rev. E 73, 036619-1 (2006).
[CrossRef]

K. Belkebir, P. C. Chaumet, and A. Sentenac, "Influence of multiple scattering on three-dimensional imaging with optical diffraction tomography," J. Opt. Soc. Am. A 23, 586-595 (2006).
[CrossRef]

2005 (1)

G. Shi and A. Nehorai, "Maximum likelihood estimation of point scatterers for computational time-reversal imaging," Commun. Inf. Syst. 5, 227-256 (2005).

2004 (3)

F. K. Gruber, E. A. Marengo, and A. J. Devaney, "Time-reversal imaging with multiple signal classification considering multiple scattering between the targets," J. Acoust. Soc. Am. 115, 3042-3047 (2004).
[CrossRef]

M. Cheney and R. Bonneau, "Imaging that exploits multipath scattering from point scatterers," Inverse Probl. 20, 1691-1711 (2004).
[CrossRef]

T. J. Cui, W. C. Chew, X. X. Yin, and W. Hong, "Study of resolution and super resolution in electromagnetic imaging for half-space problems," IEEE Trans. Antennas Propag. 52, 1398-1411 (2004).
[CrossRef]

2003 (1)

M. Bertero and P. Boccacci, "Super-resolution in computational imaging," Micron 34, 265-273 (2003).
[CrossRef] [PubMed]

2000 (1)

M. Rusek, J. Mostowski, and A. Orlowski, "Random Green matrices: From proximity resonance to Anderson localization," Phys. Rev. A 61, 022704 (2000).
[CrossRef]

1998 (4)

R. K. Snieder and J. A. Scales, "Time-reversed imaging as a diagnostic of wave and particle chaos," Phys. Rev. E 58, 5668-5675 (1998).
[CrossRef]

J. S. Hersch and E. J. Heller, "Observation of proximity resonances in a parallel-plate waveguide," Phys. Rev. Lett. 81, 3059-3062 (1998).
[CrossRef]

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

A. Kirsch, "Characterization of the shape of a scattering obstacle using the spectral data of the far field operator," Inverse Probl. 14, 1489-1512 (1998).
[CrossRef]

1996 (1)

E. J. Heller, "Quantum proximity resonance," Phys. Rev. Lett. 77, 4122-4125 (1996).
[CrossRef] [PubMed]

1994 (1)

T. Habashy and E. Wolf, "Reconstruction of scattering potentials from incomplete data," J. Mod. Opt. 41, 1679-1685 (1994).
[CrossRef]

1988 (1)

I. Tolstoy and A. Tolstoy, "Supreresonant systems of scatterers. ii," J. Acoust. Soc. Am. 83, 2086-2096 (1988).
[CrossRef]

1986 (1)

I. Tolstoy, "Supreresonant systems of scatterers. i," J. Acoust. Soc. Am. 80, 282-294 (1986).
[CrossRef]

1969 (2)

G. Toraldo Di Francia, "Degrees of freedom of an image," J. Opt. Soc. Am. 59, 799-804 (1969).
[CrossRef] [PubMed]

E. Wolf, "Three-dimensional structure determination of semi-transparent objects from holographic data," Opt. Commun. 1, 153-156 (1969).
[CrossRef]

1968 (1)

1961 (2)

D. Slepian and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty I," Bell Syst. Tech. J. 40, 43-63 (1961).

V. Twersky, "Multiple scattering of waves and optical phenomena," J. Opt. Soc. Am. 52, 145-171 (1961).
[CrossRef]

1945 (1)

L. L. Foldy, "The multiple scattering of waves. i. general theory of isotropic scattering by randomly distributed scatterers," Phys. Rev. 67, 107-119 (1945).
[CrossRef]

Appl. Phys. Lett. (2)

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

F. Simonetti, "Localization of point-like scatterers in solids with subwavelength resolution," Appl. Phys. Lett. 89, 094105 (2006).
[CrossRef]

Bell Syst. Tech. J. (1)

D. Slepian and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty I," Bell Syst. Tech. J. 40, 43-63 (1961).

Commun. Inf. Syst. (1)

G. Shi and A. Nehorai, "Maximum likelihood estimation of point scatterers for computational time-reversal imaging," Commun. Inf. Syst. 5, 227-256 (2005).

IEEE Trans. Antennas Propag. (1)

T. J. Cui, W. C. Chew, X. X. Yin, and W. Hong, "Study of resolution and super resolution in electromagnetic imaging for half-space problems," IEEE Trans. Antennas Propag. 52, 1398-1411 (2004).
[CrossRef]

IEEE Trans. Image Process. (1)

E. A. Marengo, F. K. Gruber, and F. Simonetti, "Time-reversal music imaging of extended targets," IEEE Trans. Image Process. 16, 1967-1984 (2007).
[CrossRef] [PubMed]

Inverse Probl. (4)

D. Colton and R. Kress, "Using fundamental solutions in inverse scattering," Inverse Probl. 22, R49-66 (2006).
[CrossRef]

A. Kirsch, "Characterization of the shape of a scattering obstacle using the spectral data of the far field operator," Inverse Probl. 14, 1489-1512 (1998).
[CrossRef]

S. Hou, K. Solna, and H. Zhao, "A direct imaging algorithm for extended targets," Inverse Probl. 22, 1151-1178 (2006).
[CrossRef]

M. Cheney and R. Bonneau, "Imaging that exploits multipath scattering from point scatterers," Inverse Probl. 20, 1691-1711 (2004).
[CrossRef]

J. Acoust. Soc. Am. (3)

I. Tolstoy, "Supreresonant systems of scatterers. i," J. Acoust. Soc. Am. 80, 282-294 (1986).
[CrossRef]

I. Tolstoy and A. Tolstoy, "Supreresonant systems of scatterers. ii," J. Acoust. Soc. Am. 83, 2086-2096 (1988).
[CrossRef]

F. K. Gruber, E. A. Marengo, and A. J. Devaney, "Time-reversal imaging with multiple signal classification considering multiple scattering between the targets," J. Acoust. Soc. Am. 115, 3042-3047 (2004).
[CrossRef]

J. Mod. Opt. (1)

T. Habashy and E. Wolf, "Reconstruction of scattering potentials from incomplete data," J. Mod. Opt. 41, 1679-1685 (1994).
[CrossRef]

J. Opt. Soc. Am. (3)

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

Micron (1)

M. Bertero and P. Boccacci, "Super-resolution in computational imaging," Micron 34, 265-273 (2003).
[CrossRef] [PubMed]

Opt. Commun. (1)

E. Wolf, "Three-dimensional structure determination of semi-transparent objects from holographic data," Opt. Commun. 1, 153-156 (1969).
[CrossRef]

Opt. Express (1)

Phys. Rev. (1)

L. L. Foldy, "The multiple scattering of waves. i. general theory of isotropic scattering by randomly distributed scatterers," Phys. Rev. 67, 107-119 (1945).
[CrossRef]

Phys. Rev. A (1)

M. Rusek, J. Mostowski, and A. Orlowski, "Random Green matrices: From proximity resonance to Anderson localization," Phys. Rev. A 61, 022704 (2000).
[CrossRef]

Phys. Rev. E (4)

R. K. Snieder and J. A. Scales, "Time-reversed imaging as a diagnostic of wave and particle chaos," Phys. Rev. E 58, 5668-5675 (1998).
[CrossRef]

F. Simonetti, "Multiple scattering: The key to unravel the subwavelength world from the far-field pattern of a scattered wave," Phys. Rev. E 73, 036619-1 (2006).
[CrossRef]

F. Simonetti, "Reply to 'comment on "Multiple scattering: the key to unravel the subwavelength world from the far-field pattern of a scattered wave"'," Phys. Rev. E 75, 048602 (2007).
[CrossRef]

F. Simonetti, L. Huang, N. Duric, and O. Rama, "Imaging beyond the Born approximation: An experimental investigation with an ultrasonic ring array," Phys. Rev. E 76, 036601 (2007).
[CrossRef]

Phys. Rev. Lett. (2)

E. J. Heller, "Quantum proximity resonance," Phys. Rev. Lett. 77, 4122-4125 (1996).
[CrossRef] [PubMed]

J. S. Hersch and E. J. Heller, "Observation of proximity resonances in a parallel-plate waveguide," Phys. Rev. Lett. 81, 3059-3062 (1998).
[CrossRef]

Other (8)

A. J. Devaney, "Super-resolution processing of multi-static data using time-reversal and music," unpublished; available at www.ece.neu.edu/faculty/devaney/ajd/preprints.htm (2000).

H. Lev-Ari and A. J. Devaney, "The time-reversal technique re-interpreted: subspace-based signal processing for multi-static target location," in Proceedings of the 2000 IEEE Sensor Array and Multichannel Signal Processing Workshop (IEEE, 2000), pp. 509-513.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).

E. A. Marengo and F. K. Gruber, "Subspace-based localization and inverse scattering of multiply scattering point targets," EURASIP J. Adv. Sign. Processing 2007, ID 17324 16 pp. (Hindawi Pub. Corp., 2007).

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

R. Potthast, Point Sources and Multipoles in Inverse Scattering Theory (Chapman & Hall / CRC, 2001).
[CrossRef]

J. Hadamard, Lectures on Cauchy's Problems in Linear Partial Differential Equations (Yale U. Press, 1923).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

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Figures (11)

Fig. 1
Fig. 1

For tomographic reconstructions, objects are illuminated from different directions r ̂ 0 and the scattered field is detected all around the object. (a) Diagram of a typical setup for 2D tomography; (b) two-dimensional K-space showing how the scattered field measured in the direction r ̂ and due to an incident plane wave from direction r ̂ 0 maps onto the point Ω = 2 π λ ( r ̂ 0 r ̂ ) of the K-space.

Fig. 2
Fig. 2

Solid curve, scattering cross section; dashed curve, extinction cross section, calculated under the Born approximation as a function of the distance between two scatterers relative to λ. The incident wave is parallel to the scatterers and the scattering coefficient is μ = 1.4142 + 3.4142 i ( p = 3 π 4 ) . By energy conservation the two cross sections should be the same.

Fig. 3
Fig. 3

Scattering cross section as a function of the distance between two scatterers relative to λ. The incident wave is parallel to the scatterers and the scattering coefficient is μ = 1.4142 + 3.4142 i ( p = 3 π 4 ) . Dashed curve, asymptotic Born approximation; solid curve, line Foldy–Lax model. A subwavelength resonance can be observed when d λ 0.08 .

Fig. 4
Fig. 4

Polar diagrams of the modulus of (a) the scattering amplitude and (b) its phase as a function of the observation direction ϕ. The incident wave is parallel to the scatterers ( θ = 0 ) and the scattering coefficient is μ = 1.4142 + 3.4142 i ( p = 3 π 4 ) . The separation between the scatterers is d = λ 10 . Dashed curves, Born approximation; solid curves, Foldy–Lax model.

Fig. 5
Fig. 5

Comparison between information factors estimated with and without multiple scattering when μ = 1.4142 + 3.4142 i ( p = 3 π 4 ) . The effect of multiple scattering is most evident at the proximity resonance.

Fig. 6
Fig. 6

Cross section of the TRM pseudospectrum. The scatterers are λ 10 apart and the scattering coefficient is μ = 1.4142 + 3.4142 i . Because of the absence of noise, unlimited resolution is achieved with or without multiple scattering.

Fig. 7
Fig. 7

Cross section of the TRM pseudospectrum. The scatterers are λ 10 apart and the scattering coefficient is μ = 1.4142 + 3.4142 i ( p = 3 π 4 ) . The data is corrupted with 6% random noise. Dashed curve, Born approximation; solid curve, Foldy–Lax model.

Fig. 8
Fig. 8

Performance of the TRM method for different noise levels. The scatterers are λ 10 apart and the scattering coefficient is μ = 1.4142 + 3.4142 i ( p = 3 π 4 ) . All the calculations include multiple scattering. Heavy solid curve, 10% noise; dashed curve, 20% noise; dotted curve, 30% noise; dashed-dotted curve, 40% noise; (light solid curve), 50% noise.

Fig. 9
Fig. 9

CRB (λ units) for the estimation error for the x coordinate of one of the targets and for different separation distances between scatterers. Solid curves, data contain multiple scattering; dashed curves, data are calculated with the Born approximation.

Fig. 10
Fig. 10

Same as Fig. 9 but for other different separation distances between the scatterers.

Fig. 11
Fig. 11

Relative estimation error of the distance between the two scatterers as a function of the noise level. The scatterers are λ 10 apart and the scattering coefficient is μ = 1.4142 + 3.4142 i ( p = 3 π 4 ) . The measurements are simulated with the Foldy–Lax model and corrupted with additive noise. Dashed curve, inversion model based on the Born approximation; solid curve, inversion based on the Foldy–Lax model.

Equations (41)

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ψ ( r , k r ̂ 0 ) = exp ( i k r ̂ 0 r ) D d 3 r G ( r , r ) O ( r ) ψ ( r , k r ̂ 0 ) ,
lim r ψ ( r , k r ̂ 0 ) = exp ( i k r ̂ 0 r ) + f ( k r ̂ , k r ̂ 0 ) exp ( i k r ) r ,
( T y ) ( r ̂ ) = S d s ( r ̂ 0 ) f ( k r ̂ , k r ̂ 0 ) y ( k r ̂ 0 ) .
O ̃ ( Ω ) = d 3 r O ( r ) exp ( i Ω r ) .
f ( k r ̂ , k r ̂ 0 ) = 1 4 π O ̃ [ k ( r ̂ 0 r ̂ ) ] ;
G ( r , r ) = i 4 H 0 ( k r r ) ,
r r r [ 1 ( r r ) r 2 ]
G ( r , r ) = Π e i k r r exp ( i k r ̂ r ) ,
Π = exp ( i π 4 ) 8 π k .
lim r ψ ( r , k r ̂ 0 ) = exp ( i k r ̂ 0 r ) + f ( k r ̂ , k r ̂ 0 ) exp ( i k r ) r ,
f ( k r ̂ , k r ̂ 0 ) = Π D d 2 r exp ( i k r ̂ r ) O ( r ) ψ ( r , k r ̂ 0 ) .
O ( r ) = μ δ ( r z 1 ) + μ δ ( r z 2 ) ,
μ = 2 [ exp ( i p ) + i ] ,
O ̃ ( Ω ) = 2 μ cos ( Ω z 2 ) = 2 μ cos ( Ω x d 2 ) ,
ψ s ( r , r ̂ 0 ) = μ j = 1 2 exp ( i k r ̂ 0 z j ) G ( r , z j ) ,
f BA ( ϑ , φ ) = 2 μ Π cos [ k d 2 ( cos ϑ cos φ ) ] ,
Ω x = 2 k cos ϑ ,
φ = π ϑ ,
ψ s ( r , r ̂ 0 ) = μ Π 1 μ 2 G d 2 j = 1 2 [ exp ( i k r ̂ 0 z j ) G ( r , z j ) μ G d exp ( i k r ̂ 0 z l ) G ( r , z l ) ] ,
f ( ϑ , φ ) = 2 μ Π 1 μ 2 G d 2 { cos [ k d 2 ( cos ϑ cos φ ) ] μ G d cos [ k d 2 ( cos ϑ + cos φ ) ] } .
σ = 2 k I [ μ μ 2 G d cos ( k d cos θ ) ( 1 μ G d ) ( 1 + μ G d ) ] .
f sin ( ϑ , φ ) Π = μ = O ̃ sin ( Ω ) ,
f BA x 2 μ Π [ 1 1 2 ( Ω x d 2 ) 2 ] = Π O ̃ ( Ω x , 0 ) ,
E BA x = 1 f BA x sup { f BA x Ω x } = 1 2 k d 2 ,
Ω y = 2 k sin ϑ ,
φ = ϑ .
f y = 2 μ Π 1 μ 2 G d 2 [ 1 μ G d cos ( k d 1 Ω y 2 4 k 2 ) ] ,
f x = 2 μ Π 1 μ 2 G d 2 [ cos ( Ω x d 2 ) μ G d ] .
f y 2 μ Π 1 μ 2 G d 2 [ 1 μ G d + μ G d k 2 d 2 2 ( 1 Ω y 2 4 k 2 ) ] ,
f x 2 μ Π 1 μ 2 G d 2 [ 1 μ G d 1 2 ( Ω x d 2 ) 2 ] ,
E y = 1 2 k d 2 μ G d 1 μ G d ,
E x = 1 2 k d 2 1 1 μ G d .
E y E BA x = μ G d 1 μ G d
T n = T + N ,
n = s N T ,
z ̂ 1 , z ̂ 2 = arg min z 1 , z 2 T n T ( z 1 , z 2 ) 2 ;
σ = 8 π k I [ exp ( i π 4 ) f ( r ̂ 0 , r ̂ 0 ) ] ,
f 2 = 2 π k I [ exp ( i π 4 ) f ] .
f = 1 4 π k [ 1 i 2 exp ( i p ) ] ,
ψ s = μ G .
μ = 2 [ exp ( i p ) + i ] ,

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