Abstract

This paper deals with the inverse scattering problem, in which a conducting cylinder is placed near samples that are to be reconstructed. Due to multiple scattering effect, the radius of the conducting cylinder and its distance to samples play an important role in inverse scattering problem. The paper investigates the role of the conducting cylinder under different arrangement of transmitting/receiving antennas. Numerical simulations show that with a proper arrangement of the cylinder and transmitting/receiving antennas, it is possible to achieve high-resolution reconstruction results with fewer antennas than when the conducting cylinder is absent.

© 2011 OSA

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References

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  1. A. Sentenac, C. Guerin, and P. Chaumet, “Influence of multiple scattering on the resolution of an imaging system: a Cramer-Rao analysis,” Opt. Express 15(3), 1340–1347 (2007).
    [CrossRef] [PubMed]
  2. K. Agarwal, X. Chen, and Y. Zhong, “A multipole-expansion based linear sampling method for solving inverse scattering problems,” Opt. Express 18(6), 6366–6381 (2007).
    [CrossRef]
  3. A. Sentenac, P. Chaumet, and K. Belkebir, “Beyond the Rayleigh criterion: Grating assisted far-field optical diffraction tomography,” Phys. Rev. Lett. 97(24), 243901 (2006).
    [CrossRef]
  4. J. Berezovsky, M. Borunda, and E. Heller, “Imaging coherent transport in graphene (part I): mapping universal conductance fluctuations,” Nanotechnology 21(27), 274013 (2010).
    [CrossRef] [PubMed]
  5. M. Braun, L. Chirolli, and G. Burkard, “Signature of chirality in scanning-probe imaging of charge flow in graphene,” Phys. Rev. B 77(11), 115433 (2008).
    [CrossRef]
  6. J. Li, H. Liu, and J. Zou, “Strengthened linear sampling method with a reference ball,” SIAM J. Sci. Comput. 31(6), 4013–4040 (2009).
    [CrossRef]
  7. X. Chen, “MUSIC imaging applied to total internal reflection tomography,” J. Opt. Soc. Am. A 25(2), 357–364 (2008).
    [CrossRef]
  8. P. Carney and J. Schotland, “Theory of total-internal reflection tomography,” J. Opt. Soc. Am. A 20(3), 542–547 (2003).
    [CrossRef]
  9. X. Chen and Y. Zhong, “Influence of multiple scattering on subwavelength imaging: transverse electric case,” J. Opt. Soc. Am. A 27(2), 245–250 (2010).
    [CrossRef]
  10. K. Belkebir, P. C. Chaumet, and A. Sentenac, “Superresolution in total internal reflection tomography,” J. Opt. Soc. Am. A 22(9), 1889–1897 (2005).
    [CrossRef]
  11. L. Crocco, M. D’Urso, and T. Isernia, “Inverse scattering from phaseless measurements of the total field on a closed curve,” J. Opt. Soc. Am. A 21(4), 622–631 (2004).
    [CrossRef]
  12. O. Bucci, L. Crocco, M. D’Urso, and T. Isernia, “Inverse scattering from phaseless measurements of the total field on open lines,” J. Opt. Soc. Am. A 23(10), 2566–2577 (2006).
    [CrossRef]
  13. L. Crocco, M. D’Urso, and T. Isernia, “Faithful non-linear imaging from only-amplitude measurements of incident and total fields,” Opt. Express 15(7), 3804–3815 (2007).
    [CrossRef] [PubMed]
  14. W. Chew and Y. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method,” IEEE Trans. Med. Imaging 9(2), 218–225 (1990).
    [CrossRef] [PubMed]
  15. S. Norton, “Iterative inverse scattering algorithms: Methods of computing Frechet derivatives,” J. Acoust. Soc. Am. 106(5), 2653–2660 (1999).
    [CrossRef]
  16. X. Chen, “Application of signal-subspace and optimization methods in reconstructing extended scatterers,” J. Opt. Soc. Am. A 26(4), 1022–1026 (2009).
    [CrossRef]
  17. X. Chen, “Subspace-based optimization method for solving inverse scattering problems,” IEEE Trans. Geosci. Remote Sens. 48(1), 42–49 (2010).
    [CrossRef]
  18. Y. Zhong and X. Chen, “Twofold subspace-based optimization method for solving inverse scattering problems,” Inverse Probl. 25(8), 085003 (2009).
    [CrossRef]
  19. J. Richmond, “Scattering by a dielectric cylinder of arbitrary cross section shape,” IEEE Trans. Antennas Propag. 13(3), 334–341 (1965).
    [CrossRef]
  20. L. Pan, X. Chen, Y. Zhong, and S. P. Yeo, “Comparison among the variants of subspace-based optimization method for addressing inverse scattering problems: transverse electric case,” J. Opt. Soc. Am. A 27(10), 2208–2215 (2010).
    [CrossRef]
  21. K. Agarwal, L. Pan, and X. Chen, “Subspace-Based Optimization Method for Reconstruction of 2-D Complex Anisotropic Dielectric Objects,” IEEE Trans. Microwave Theory Tech. 58(4), 1065–1074 (2010).
    [CrossRef]
  22. X. Ye, X. Chen, Y. Zhong, and K. Agarwal, “Subspace-based optimization method for reconstructing perfectly electric conductors,” Prog. Electromagn. Res. 100, 119–128 (2010).
    [CrossRef]
  23. Y. Zhong, X. Chen, and K. Agarwal, “An improved subspace-based optimization method and its implementation in solving three-dimensional inverse problems,” IEEE Trans. Geosci. Remote Sens. 48(10), 3763–3768 (2010).
    [CrossRef]
  24. Y. Zhong and X. Chen, “MUSIC imaging and electromagneitc inverse scattering of multiple-scattering small anisotropic spheres,” IEEE Trans. Antenn. Propag. 55(12), 3542–3549 (2007).
    [CrossRef]

2010

J. Berezovsky, M. Borunda, and E. Heller, “Imaging coherent transport in graphene (part I): mapping universal conductance fluctuations,” Nanotechnology 21(27), 274013 (2010).
[CrossRef] [PubMed]

X. Chen and Y. Zhong, “Influence of multiple scattering on subwavelength imaging: transverse electric case,” J. Opt. Soc. Am. A 27(2), 245–250 (2010).
[CrossRef]

L. Pan, X. Chen, Y. Zhong, and S. P. Yeo, “Comparison among the variants of subspace-based optimization method for addressing inverse scattering problems: transverse electric case,” J. Opt. Soc. Am. A 27(10), 2208–2215 (2010).
[CrossRef]

K. Agarwal, L. Pan, and X. Chen, “Subspace-Based Optimization Method for Reconstruction of 2-D Complex Anisotropic Dielectric Objects,” IEEE Trans. Microwave Theory Tech. 58(4), 1065–1074 (2010).
[CrossRef]

X. Ye, X. Chen, Y. Zhong, and K. Agarwal, “Subspace-based optimization method for reconstructing perfectly electric conductors,” Prog. Electromagn. Res. 100, 119–128 (2010).
[CrossRef]

Y. Zhong, X. Chen, and K. Agarwal, “An improved subspace-based optimization method and its implementation in solving three-dimensional inverse problems,” IEEE Trans. Geosci. Remote Sens. 48(10), 3763–3768 (2010).
[CrossRef]

X. Chen, “Subspace-based optimization method for solving inverse scattering problems,” IEEE Trans. Geosci. Remote Sens. 48(1), 42–49 (2010).
[CrossRef]

2009

Y. Zhong and X. Chen, “Twofold subspace-based optimization method for solving inverse scattering problems,” Inverse Probl. 25(8), 085003 (2009).
[CrossRef]

X. Chen, “Application of signal-subspace and optimization methods in reconstructing extended scatterers,” J. Opt. Soc. Am. A 26(4), 1022–1026 (2009).
[CrossRef]

J. Li, H. Liu, and J. Zou, “Strengthened linear sampling method with a reference ball,” SIAM J. Sci. Comput. 31(6), 4013–4040 (2009).
[CrossRef]

2008

X. Chen, “MUSIC imaging applied to total internal reflection tomography,” J. Opt. Soc. Am. A 25(2), 357–364 (2008).
[CrossRef]

M. Braun, L. Chirolli, and G. Burkard, “Signature of chirality in scanning-probe imaging of charge flow in graphene,” Phys. Rev. B 77(11), 115433 (2008).
[CrossRef]

2007

2006

O. Bucci, L. Crocco, M. D’Urso, and T. Isernia, “Inverse scattering from phaseless measurements of the total field on open lines,” J. Opt. Soc. Am. A 23(10), 2566–2577 (2006).
[CrossRef]

A. Sentenac, P. Chaumet, and K. Belkebir, “Beyond the Rayleigh criterion: Grating assisted far-field optical diffraction tomography,” Phys. Rev. Lett. 97(24), 243901 (2006).
[CrossRef]

2005

2004

2003

1999

S. Norton, “Iterative inverse scattering algorithms: Methods of computing Frechet derivatives,” J. Acoust. Soc. Am. 106(5), 2653–2660 (1999).
[CrossRef]

1990

W. Chew and Y. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method,” IEEE Trans. Med. Imaging 9(2), 218–225 (1990).
[CrossRef] [PubMed]

1965

J. Richmond, “Scattering by a dielectric cylinder of arbitrary cross section shape,” IEEE Trans. Antennas Propag. 13(3), 334–341 (1965).
[CrossRef]

Agarwal, K.

K. Agarwal, L. Pan, and X. Chen, “Subspace-Based Optimization Method for Reconstruction of 2-D Complex Anisotropic Dielectric Objects,” IEEE Trans. Microwave Theory Tech. 58(4), 1065–1074 (2010).
[CrossRef]

X. Ye, X. Chen, Y. Zhong, and K. Agarwal, “Subspace-based optimization method for reconstructing perfectly electric conductors,” Prog. Electromagn. Res. 100, 119–128 (2010).
[CrossRef]

Y. Zhong, X. Chen, and K. Agarwal, “An improved subspace-based optimization method and its implementation in solving three-dimensional inverse problems,” IEEE Trans. Geosci. Remote Sens. 48(10), 3763–3768 (2010).
[CrossRef]

K. Agarwal, X. Chen, and Y. Zhong, “A multipole-expansion based linear sampling method for solving inverse scattering problems,” Opt. Express 18(6), 6366–6381 (2007).
[CrossRef]

Belkebir, K.

A. Sentenac, P. Chaumet, and K. Belkebir, “Beyond the Rayleigh criterion: Grating assisted far-field optical diffraction tomography,” Phys. Rev. Lett. 97(24), 243901 (2006).
[CrossRef]

K. Belkebir, P. C. Chaumet, and A. Sentenac, “Superresolution in total internal reflection tomography,” J. Opt. Soc. Am. A 22(9), 1889–1897 (2005).
[CrossRef]

Berezovsky, J.

J. Berezovsky, M. Borunda, and E. Heller, “Imaging coherent transport in graphene (part I): mapping universal conductance fluctuations,” Nanotechnology 21(27), 274013 (2010).
[CrossRef] [PubMed]

Borunda, M.

J. Berezovsky, M. Borunda, and E. Heller, “Imaging coherent transport in graphene (part I): mapping universal conductance fluctuations,” Nanotechnology 21(27), 274013 (2010).
[CrossRef] [PubMed]

Braun, M.

M. Braun, L. Chirolli, and G. Burkard, “Signature of chirality in scanning-probe imaging of charge flow in graphene,” Phys. Rev. B 77(11), 115433 (2008).
[CrossRef]

Bucci, O.

Burkard, G.

M. Braun, L. Chirolli, and G. Burkard, “Signature of chirality in scanning-probe imaging of charge flow in graphene,” Phys. Rev. B 77(11), 115433 (2008).
[CrossRef]

Carney, P.

Chaumet, P.

A. Sentenac, C. Guerin, and P. Chaumet, “Influence of multiple scattering on the resolution of an imaging system: a Cramer-Rao analysis,” Opt. Express 15(3), 1340–1347 (2007).
[CrossRef] [PubMed]

A. Sentenac, P. Chaumet, and K. Belkebir, “Beyond the Rayleigh criterion: Grating assisted far-field optical diffraction tomography,” Phys. Rev. Lett. 97(24), 243901 (2006).
[CrossRef]

Chaumet, P. C.

Chen, X.

X. Chen and Y. Zhong, “Influence of multiple scattering on subwavelength imaging: transverse electric case,” J. Opt. Soc. Am. A 27(2), 245–250 (2010).
[CrossRef]

Y. Zhong, X. Chen, and K. Agarwal, “An improved subspace-based optimization method and its implementation in solving three-dimensional inverse problems,” IEEE Trans. Geosci. Remote Sens. 48(10), 3763–3768 (2010).
[CrossRef]

X. Ye, X. Chen, Y. Zhong, and K. Agarwal, “Subspace-based optimization method for reconstructing perfectly electric conductors,” Prog. Electromagn. Res. 100, 119–128 (2010).
[CrossRef]

K. Agarwal, L. Pan, and X. Chen, “Subspace-Based Optimization Method for Reconstruction of 2-D Complex Anisotropic Dielectric Objects,” IEEE Trans. Microwave Theory Tech. 58(4), 1065–1074 (2010).
[CrossRef]

L. Pan, X. Chen, Y. Zhong, and S. P. Yeo, “Comparison among the variants of subspace-based optimization method for addressing inverse scattering problems: transverse electric case,” J. Opt. Soc. Am. A 27(10), 2208–2215 (2010).
[CrossRef]

X. Chen, “Subspace-based optimization method for solving inverse scattering problems,” IEEE Trans. Geosci. Remote Sens. 48(1), 42–49 (2010).
[CrossRef]

Y. Zhong and X. Chen, “Twofold subspace-based optimization method for solving inverse scattering problems,” Inverse Probl. 25(8), 085003 (2009).
[CrossRef]

X. Chen, “Application of signal-subspace and optimization methods in reconstructing extended scatterers,” J. Opt. Soc. Am. A 26(4), 1022–1026 (2009).
[CrossRef]

X. Chen, “MUSIC imaging applied to total internal reflection tomography,” J. Opt. Soc. Am. A 25(2), 357–364 (2008).
[CrossRef]

K. Agarwal, X. Chen, and Y. Zhong, “A multipole-expansion based linear sampling method for solving inverse scattering problems,” Opt. Express 18(6), 6366–6381 (2007).
[CrossRef]

Y. Zhong and X. Chen, “MUSIC imaging and electromagneitc inverse scattering of multiple-scattering small anisotropic spheres,” IEEE Trans. Antenn. Propag. 55(12), 3542–3549 (2007).
[CrossRef]

Chew, W.

W. Chew and Y. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method,” IEEE Trans. Med. Imaging 9(2), 218–225 (1990).
[CrossRef] [PubMed]

Chirolli, L.

M. Braun, L. Chirolli, and G. Burkard, “Signature of chirality in scanning-probe imaging of charge flow in graphene,” Phys. Rev. B 77(11), 115433 (2008).
[CrossRef]

Crocco, L.

D’Urso, M.

Guerin, C.

Heller, E.

J. Berezovsky, M. Borunda, and E. Heller, “Imaging coherent transport in graphene (part I): mapping universal conductance fluctuations,” Nanotechnology 21(27), 274013 (2010).
[CrossRef] [PubMed]

Isernia, T.

Li, J.

J. Li, H. Liu, and J. Zou, “Strengthened linear sampling method with a reference ball,” SIAM J. Sci. Comput. 31(6), 4013–4040 (2009).
[CrossRef]

Liu, H.

J. Li, H. Liu, and J. Zou, “Strengthened linear sampling method with a reference ball,” SIAM J. Sci. Comput. 31(6), 4013–4040 (2009).
[CrossRef]

Norton, S.

S. Norton, “Iterative inverse scattering algorithms: Methods of computing Frechet derivatives,” J. Acoust. Soc. Am. 106(5), 2653–2660 (1999).
[CrossRef]

Pan, L.

L. Pan, X. Chen, Y. Zhong, and S. P. Yeo, “Comparison among the variants of subspace-based optimization method for addressing inverse scattering problems: transverse electric case,” J. Opt. Soc. Am. A 27(10), 2208–2215 (2010).
[CrossRef]

K. Agarwal, L. Pan, and X. Chen, “Subspace-Based Optimization Method for Reconstruction of 2-D Complex Anisotropic Dielectric Objects,” IEEE Trans. Microwave Theory Tech. 58(4), 1065–1074 (2010).
[CrossRef]

Richmond, J.

J. Richmond, “Scattering by a dielectric cylinder of arbitrary cross section shape,” IEEE Trans. Antennas Propag. 13(3), 334–341 (1965).
[CrossRef]

Schotland, J.

Sentenac, A.

Wang, Y.

W. Chew and Y. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method,” IEEE Trans. Med. Imaging 9(2), 218–225 (1990).
[CrossRef] [PubMed]

Ye, X.

X. Ye, X. Chen, Y. Zhong, and K. Agarwal, “Subspace-based optimization method for reconstructing perfectly electric conductors,” Prog. Electromagn. Res. 100, 119–128 (2010).
[CrossRef]

Yeo, S. P.

Zhong, Y.

L. Pan, X. Chen, Y. Zhong, and S. P. Yeo, “Comparison among the variants of subspace-based optimization method for addressing inverse scattering problems: transverse electric case,” J. Opt. Soc. Am. A 27(10), 2208–2215 (2010).
[CrossRef]

X. Ye, X. Chen, Y. Zhong, and K. Agarwal, “Subspace-based optimization method for reconstructing perfectly electric conductors,” Prog. Electromagn. Res. 100, 119–128 (2010).
[CrossRef]

Y. Zhong, X. Chen, and K. Agarwal, “An improved subspace-based optimization method and its implementation in solving three-dimensional inverse problems,” IEEE Trans. Geosci. Remote Sens. 48(10), 3763–3768 (2010).
[CrossRef]

X. Chen and Y. Zhong, “Influence of multiple scattering on subwavelength imaging: transverse electric case,” J. Opt. Soc. Am. A 27(2), 245–250 (2010).
[CrossRef]

Y. Zhong and X. Chen, “Twofold subspace-based optimization method for solving inverse scattering problems,” Inverse Probl. 25(8), 085003 (2009).
[CrossRef]

Y. Zhong and X. Chen, “MUSIC imaging and electromagneitc inverse scattering of multiple-scattering small anisotropic spheres,” IEEE Trans. Antenn. Propag. 55(12), 3542–3549 (2007).
[CrossRef]

K. Agarwal, X. Chen, and Y. Zhong, “A multipole-expansion based linear sampling method for solving inverse scattering problems,” Opt. Express 18(6), 6366–6381 (2007).
[CrossRef]

Zou, J.

J. Li, H. Liu, and J. Zou, “Strengthened linear sampling method with a reference ball,” SIAM J. Sci. Comput. 31(6), 4013–4040 (2009).
[CrossRef]

IEEE Trans. Antenn. Propag.

Y. Zhong and X. Chen, “MUSIC imaging and electromagneitc inverse scattering of multiple-scattering small anisotropic spheres,” IEEE Trans. Antenn. Propag. 55(12), 3542–3549 (2007).
[CrossRef]

IEEE Trans. Antennas Propag.

J. Richmond, “Scattering by a dielectric cylinder of arbitrary cross section shape,” IEEE Trans. Antennas Propag. 13(3), 334–341 (1965).
[CrossRef]

IEEE Trans. Geosci. Remote Sens.

X. Chen, “Subspace-based optimization method for solving inverse scattering problems,” IEEE Trans. Geosci. Remote Sens. 48(1), 42–49 (2010).
[CrossRef]

Y. Zhong, X. Chen, and K. Agarwal, “An improved subspace-based optimization method and its implementation in solving three-dimensional inverse problems,” IEEE Trans. Geosci. Remote Sens. 48(10), 3763–3768 (2010).
[CrossRef]

IEEE Trans. Med. Imaging

W. Chew and Y. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method,” IEEE Trans. Med. Imaging 9(2), 218–225 (1990).
[CrossRef] [PubMed]

IEEE Trans. Microwave Theory Tech.

K. Agarwal, L. Pan, and X. Chen, “Subspace-Based Optimization Method for Reconstruction of 2-D Complex Anisotropic Dielectric Objects,” IEEE Trans. Microwave Theory Tech. 58(4), 1065–1074 (2010).
[CrossRef]

Inverse Probl.

Y. Zhong and X. Chen, “Twofold subspace-based optimization method for solving inverse scattering problems,” Inverse Probl. 25(8), 085003 (2009).
[CrossRef]

J. Acoust. Soc. Am.

S. Norton, “Iterative inverse scattering algorithms: Methods of computing Frechet derivatives,” J. Acoust. Soc. Am. 106(5), 2653–2660 (1999).
[CrossRef]

J. Opt. Soc. Am. A

Nanotechnology

J. Berezovsky, M. Borunda, and E. Heller, “Imaging coherent transport in graphene (part I): mapping universal conductance fluctuations,” Nanotechnology 21(27), 274013 (2010).
[CrossRef] [PubMed]

Opt. Express

Phys. Rev. B

M. Braun, L. Chirolli, and G. Burkard, “Signature of chirality in scanning-probe imaging of charge flow in graphene,” Phys. Rev. B 77(11), 115433 (2008).
[CrossRef]

Phys. Rev. Lett.

A. Sentenac, P. Chaumet, and K. Belkebir, “Beyond the Rayleigh criterion: Grating assisted far-field optical diffraction tomography,” Phys. Rev. Lett. 97(24), 243901 (2006).
[CrossRef]

Prog. Electromagn. Res.

X. Ye, X. Chen, Y. Zhong, and K. Agarwal, “Subspace-based optimization method for reconstructing perfectly electric conductors,” Prog. Electromagn. Res. 100, 119–128 (2010).
[CrossRef]

SIAM J. Sci. Comput.

J. Li, H. Liu, and J. Zou, “Strengthened linear sampling method with a reference ball,” SIAM J. Sci. Comput. 31(6), 4013–4040 (2009).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

A view to the configuration with the test domain, the conducting cylinder and the distribution of the sources and the receivers.

Fig. 2
Fig. 2

The ’Austria’ profile adopted for the numerical simulations.

Fig. 3
Fig. 3

The simulated errors of the reconstruction results for the combinations described in case I.

Fig. 4
Fig. 4

The simulation results when the antennas are distributed as described in case I. (a) is for the combination of R = 0.5λ,r = 3.15λ and (b) is for the case with no cylinder.

Fig. 5
Fig. 5

The simulated errors of the reconstruction results for the combinations described in case II.

Fig. 6
Fig. 6

The simulation results when the antennas are distributed as described in case II. (a) is for the combination of R = 0.3λ,r = 2.1λ and (b) is for the case with no cylinder.

Fig. 7
Fig. 7

The simulated errors of the reconstruction results for the combinations described in case III.

Fig. 8
Fig. 8

The simulation results when the antennas are distributed as described in case III. (a) is for the combination of R = 0.5λ,r = 2.1λ and (b) is for the case with no cylinder.

Fig. 9
Fig. 9

The simulation results of new scatterers when the antennas are distributed as described in case III. (a) is for the combination of R = 0.3λ,r = 2.1λ and (b) is for the case with no cylinder.

Equations (11)

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

E ¯ d , p inc ( r ¯ m , r ¯ p i ) = k η 0 4 H 0 ( 1 ) ( k | r ¯ m r ¯ p i | ) ,
E ¯ s , p inc ( r ¯ m , r ¯ p i ) = k η 0 4 l = + S l H l ( 1 ) ( k ρ p i ) e i l θ p i H l ( 1 ) ( k ρ m ) e i l θ m ,
I ¯ p d ( r ¯ m ) = i ω ε 0 [ ε r ( r ¯ m ) 1 ] E ¯ p tot ( r ¯ m ) ,
E ¯ p tot ( r ¯ m ) = E ¯ p inc ( r ¯ m ) + E ¯ d , p sca ( r ¯ m ) + E ¯ s , p sca ( r ¯ m ) ,
E ¯ d , p sca ( r ¯ m ) = k η 0 4 n = 1 M cell n H 0 ( 1 ) ( k | r ¯ n r ¯ m | ) I ¯ p d ( r ¯ n ) d σ ,
E ¯ s , p sca ( r ¯ m ) = k η 0 4 n = 1 M l = + S l cell n H l ( 1 ) ( k ρ n ) e i l θ n I ¯ p d ( r ¯ n ) d σ H l ( 1 ) ( k ρ m ) e i l θ m .
E ¯ p sca ( r ¯ q ) = E ¯ d , p sca ( r ¯ q ) + E ¯ s , p sca ( r ¯ q ) .
I ¯ p d = ξ ¯ ¯ · ( E ¯ p inc + G ¯ ¯ D · I ¯ p d ) = ξ ¯ ¯ · [ E ¯ p inc + ( g ¯ ¯ D + h ¯ ¯ D ) · I ¯ p d ] ,
E ¯ p sca = G ¯ ¯ S · I ¯ p d = ( g ¯ ¯ S + h ¯ ¯ S ) · I ¯ p d
f ( α ¯ 1 n , α ¯ 2 n , , α ¯ N i n , ξ ¯ ¯ ) = p = 1 N i ( G ¯ ¯ S · V ¯ ¯ n · α ¯ p n + G ¯ ¯ S · I ¯ p s E ¯ p sca 2 E ¯ p sca 2 + A ¯ ¯ · α ¯ p n B ¯ p 2 I ¯ p s 2 )
E 2 D = 1 n u m m = 1 64 n = 1 64 | ε ¯ ¯ m , n act ε ¯ ¯ m , n sim ε ¯ m , n act | 2 ,

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