Abstract

It has recently been shown that the measurement setups usually adopted in inverse scattering problems, in which the primary sources and the receiving antennas are placed at some wavelength apart from the object under test, suffer from intrinsic limitations in the reconstruction capabilities because of the essentially finite-dimensional nature of the space of data (the scattered fields). To investigate whether it is possible to overcome these limitations, two (novel to our knowledge) measurement configurations for inverse scattering experiments at a fixed frequency are analyzed and discussed. By means of an analysis of the properties of the radiation operator, it is shown that positioning the measurement probes (and possibly the primary sources) in the close proximity of the object under test allows an improvement of the reconstruction capabilities of inversion algorithms with respect to conventional setups. However, such an improvement can be achieved only in a region close to the border of the region under test. Quantitative rules for the achievable improvement are given and are exemplified through numerical examples.

© 1999 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. C. Bolomey, “Recent European developments in active microwave imaging for industrial, scientific, and medical applications,” IEEE Trans. Microwave Theory Tech. 37, 2109–2117 (1991).
    [CrossRef]
  2. P. M. Meaney, K. D. Paulsen, J. T. Chang, “Near-field microwave imaging of biologically-based materials using a monopole transceiver system,” IEEE Trans. Microwave Theory Tech. 46, 31–45 (1998).
    [CrossRef]
  3. D. J. Daniels, “Surface penetrating radar,” Electron. Commun. Eng. J. 8, 165–182 (1996).
    [CrossRef]
  4. M. Bertero, “Linear inverse and ill-posed problems,” Adv. Electron. Electron Phys. 75, 1–120 (1989).
    [CrossRef]
  5. T. Isernia, V. Pascazio, R. Pierri, “A nonlinear estimation method in tomographic imaging,” IEEE Trans. Geosci. Remote Sens. 35, 910–923 (1997).
    [CrossRef]
  6. R. Pierri, A. Brancaccio, “Imaging of a rotationally symmetric dielectric cylinder by a quadratic approach,” J. Opt. Soc. Am. A 14, 2777–2785 (1997).
    [CrossRef]
  7. W. C. Chew, Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method,” IEEE Trans. Med. Imaging 9, 218–225 (1990).
    [CrossRef] [PubMed]
  8. T. M. Habashy, R. W. Groom, B. P. Spies, “Beyond the Born and Rytov approximations: a nonlinear approach to electromagnetic scattering,” J. Geophys. Res. 98, 1759–1775 (1993).
    [CrossRef]
  9. O. M. Bucci, T. Isernia, “Electromagnetic inverse scattering: retrievable information and measurement strategies,” Radio Sci. 32, 2123–2138 (1997).
    [CrossRef]
  10. W. Liang, “A probe for making near-field measurement with minimal disturbance: the optically modulated scatterer,” IEEE Trans. Antennas Propag. 45, 772–779 (1997).
    [CrossRef]
  11. G. Breglio, O. M. Bucci, A. Cutolo, R. Massa, G. Panariello, “Electro-optical sensing of GHz electromagnetic fields,” in Proceedings of International Conference on Electromagnetics in Advanced Applications 1997 (Politecnico di Torino, Torino, Italy, 1997), pp. 499–502.
  12. F. N. Kong, T. L. By, “Performance of a GPR system which uses step frequency signals,” J. Appl. Geophys. 33, 15–26 (1995).
    [CrossRef]
  13. R. P. Porter, A. J. Devaney, “Generalized holography and computational solutions to inverse source problems,” J. Opt. Soc. Am. 72, 1707–1713 (1982).
    [CrossRef]
  14. O. M. Bucci, G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antennas Propag. 37, 918–926 (1989).
    [CrossRef]
  15. T. Isernia, G. Leone, R. Pierri, F. Soldovieri, “On the local minima problem in phase reconstruction algorithms,” Radio Sci. 31, 1887–1899 (1996).
    [CrossRef]
  16. M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
  17. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products, A. Jeffrey, ed. (Academic, London, 1997).
  18. W. C. Chew, Y. M. Wang, G. Otto, D. Lesselier, J. Ch. Bolomey, “On the inverse source method of solving inverse scattering problems,” Inverse Probl. 10, 547–553 (1994).
    [CrossRef]
  19. This is equivalent to assuming that the actual field in a given point depends mainly on the incident field in that point, or, alternatively, that the value of the incident field in a given point affects only values of E in the points nearby.
  20. W. C. Chew, J. H. Lin, “A frequency-hopping approach for microwave imaging of large inhomogeneous bodies,” IEEE Microwave Guid. Wave Lett. 5, 439–441 (1995).
    [CrossRef]

1998 (1)

P. M. Meaney, K. D. Paulsen, J. T. Chang, “Near-field microwave imaging of biologically-based materials using a monopole transceiver system,” IEEE Trans. Microwave Theory Tech. 46, 31–45 (1998).
[CrossRef]

1997 (4)

T. Isernia, V. Pascazio, R. Pierri, “A nonlinear estimation method in tomographic imaging,” IEEE Trans. Geosci. Remote Sens. 35, 910–923 (1997).
[CrossRef]

O. M. Bucci, T. Isernia, “Electromagnetic inverse scattering: retrievable information and measurement strategies,” Radio Sci. 32, 2123–2138 (1997).
[CrossRef]

W. Liang, “A probe for making near-field measurement with minimal disturbance: the optically modulated scatterer,” IEEE Trans. Antennas Propag. 45, 772–779 (1997).
[CrossRef]

R. Pierri, A. Brancaccio, “Imaging of a rotationally symmetric dielectric cylinder by a quadratic approach,” J. Opt. Soc. Am. A 14, 2777–2785 (1997).
[CrossRef]

1996 (2)

T. Isernia, G. Leone, R. Pierri, F. Soldovieri, “On the local minima problem in phase reconstruction algorithms,” Radio Sci. 31, 1887–1899 (1996).
[CrossRef]

D. J. Daniels, “Surface penetrating radar,” Electron. Commun. Eng. J. 8, 165–182 (1996).
[CrossRef]

1995 (2)

F. N. Kong, T. L. By, “Performance of a GPR system which uses step frequency signals,” J. Appl. Geophys. 33, 15–26 (1995).
[CrossRef]

W. C. Chew, J. H. Lin, “A frequency-hopping approach for microwave imaging of large inhomogeneous bodies,” IEEE Microwave Guid. Wave Lett. 5, 439–441 (1995).
[CrossRef]

1994 (1)

W. C. Chew, Y. M. Wang, G. Otto, D. Lesselier, J. Ch. Bolomey, “On the inverse source method of solving inverse scattering problems,” Inverse Probl. 10, 547–553 (1994).
[CrossRef]

1993 (1)

T. M. Habashy, R. W. Groom, B. P. Spies, “Beyond the Born and Rytov approximations: a nonlinear approach to electromagnetic scattering,” J. Geophys. Res. 98, 1759–1775 (1993).
[CrossRef]

1991 (1)

J. C. Bolomey, “Recent European developments in active microwave imaging for industrial, scientific, and medical applications,” IEEE Trans. Microwave Theory Tech. 37, 2109–2117 (1991).
[CrossRef]

1990 (1)

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

1989 (2)

M. Bertero, “Linear inverse and ill-posed problems,” Adv. Electron. Electron Phys. 75, 1–120 (1989).
[CrossRef]

O. M. Bucci, G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antennas Propag. 37, 918–926 (1989).
[CrossRef]

1982 (1)

Abramowitz, M.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Bertero, M.

M. Bertero, “Linear inverse and ill-posed problems,” Adv. Electron. Electron Phys. 75, 1–120 (1989).
[CrossRef]

Bolomey, J. C.

J. C. Bolomey, “Recent European developments in active microwave imaging for industrial, scientific, and medical applications,” IEEE Trans. Microwave Theory Tech. 37, 2109–2117 (1991).
[CrossRef]

Bolomey, J. Ch.

W. C. Chew, Y. M. Wang, G. Otto, D. Lesselier, J. Ch. Bolomey, “On the inverse source method of solving inverse scattering problems,” Inverse Probl. 10, 547–553 (1994).
[CrossRef]

Brancaccio, A.

Breglio, G.

G. Breglio, O. M. Bucci, A. Cutolo, R. Massa, G. Panariello, “Electro-optical sensing of GHz electromagnetic fields,” in Proceedings of International Conference on Electromagnetics in Advanced Applications 1997 (Politecnico di Torino, Torino, Italy, 1997), pp. 499–502.

Bucci, O. M.

O. M. Bucci, T. Isernia, “Electromagnetic inverse scattering: retrievable information and measurement strategies,” Radio Sci. 32, 2123–2138 (1997).
[CrossRef]

O. M. Bucci, G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antennas Propag. 37, 918–926 (1989).
[CrossRef]

G. Breglio, O. M. Bucci, A. Cutolo, R. Massa, G. Panariello, “Electro-optical sensing of GHz electromagnetic fields,” in Proceedings of International Conference on Electromagnetics in Advanced Applications 1997 (Politecnico di Torino, Torino, Italy, 1997), pp. 499–502.

By, T. L.

F. N. Kong, T. L. By, “Performance of a GPR system which uses step frequency signals,” J. Appl. Geophys. 33, 15–26 (1995).
[CrossRef]

Chang, J. T.

P. M. Meaney, K. D. Paulsen, J. T. Chang, “Near-field microwave imaging of biologically-based materials using a monopole transceiver system,” IEEE Trans. Microwave Theory Tech. 46, 31–45 (1998).
[CrossRef]

Chew, W. C.

W. C. Chew, J. H. Lin, “A frequency-hopping approach for microwave imaging of large inhomogeneous bodies,” IEEE Microwave Guid. Wave Lett. 5, 439–441 (1995).
[CrossRef]

W. C. Chew, Y. M. Wang, G. Otto, D. Lesselier, J. Ch. Bolomey, “On the inverse source method of solving inverse scattering problems,” Inverse Probl. 10, 547–553 (1994).
[CrossRef]

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

Cutolo, A.

G. Breglio, O. M. Bucci, A. Cutolo, R. Massa, G. Panariello, “Electro-optical sensing of GHz electromagnetic fields,” in Proceedings of International Conference on Electromagnetics in Advanced Applications 1997 (Politecnico di Torino, Torino, Italy, 1997), pp. 499–502.

Daniels, D. J.

D. J. Daniels, “Surface penetrating radar,” Electron. Commun. Eng. J. 8, 165–182 (1996).
[CrossRef]

Devaney, A. J.

Franceschetti, G.

O. M. Bucci, G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antennas Propag. 37, 918–926 (1989).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products, A. Jeffrey, ed. (Academic, London, 1997).

Groom, R. W.

T. M. Habashy, R. W. Groom, B. P. Spies, “Beyond the Born and Rytov approximations: a nonlinear approach to electromagnetic scattering,” J. Geophys. Res. 98, 1759–1775 (1993).
[CrossRef]

Habashy, T. M.

T. M. Habashy, R. W. Groom, B. P. Spies, “Beyond the Born and Rytov approximations: a nonlinear approach to electromagnetic scattering,” J. Geophys. Res. 98, 1759–1775 (1993).
[CrossRef]

Isernia, T.

O. M. Bucci, T. Isernia, “Electromagnetic inverse scattering: retrievable information and measurement strategies,” Radio Sci. 32, 2123–2138 (1997).
[CrossRef]

T. Isernia, V. Pascazio, R. Pierri, “A nonlinear estimation method in tomographic imaging,” IEEE Trans. Geosci. Remote Sens. 35, 910–923 (1997).
[CrossRef]

T. Isernia, G. Leone, R. Pierri, F. Soldovieri, “On the local minima problem in phase reconstruction algorithms,” Radio Sci. 31, 1887–1899 (1996).
[CrossRef]

Kong, F. N.

F. N. Kong, T. L. By, “Performance of a GPR system which uses step frequency signals,” J. Appl. Geophys. 33, 15–26 (1995).
[CrossRef]

Leone, G.

T. Isernia, G. Leone, R. Pierri, F. Soldovieri, “On the local minima problem in phase reconstruction algorithms,” Radio Sci. 31, 1887–1899 (1996).
[CrossRef]

Lesselier, D.

W. C. Chew, Y. M. Wang, G. Otto, D. Lesselier, J. Ch. Bolomey, “On the inverse source method of solving inverse scattering problems,” Inverse Probl. 10, 547–553 (1994).
[CrossRef]

Liang, W.

W. Liang, “A probe for making near-field measurement with minimal disturbance: the optically modulated scatterer,” IEEE Trans. Antennas Propag. 45, 772–779 (1997).
[CrossRef]

Lin, J. H.

W. C. Chew, J. H. Lin, “A frequency-hopping approach for microwave imaging of large inhomogeneous bodies,” IEEE Microwave Guid. Wave Lett. 5, 439–441 (1995).
[CrossRef]

Massa, R.

G. Breglio, O. M. Bucci, A. Cutolo, R. Massa, G. Panariello, “Electro-optical sensing of GHz electromagnetic fields,” in Proceedings of International Conference on Electromagnetics in Advanced Applications 1997 (Politecnico di Torino, Torino, Italy, 1997), pp. 499–502.

Meaney, P. M.

P. M. Meaney, K. D. Paulsen, J. T. Chang, “Near-field microwave imaging of biologically-based materials using a monopole transceiver system,” IEEE Trans. Microwave Theory Tech. 46, 31–45 (1998).
[CrossRef]

Otto, G.

W. C. Chew, Y. M. Wang, G. Otto, D. Lesselier, J. Ch. Bolomey, “On the inverse source method of solving inverse scattering problems,” Inverse Probl. 10, 547–553 (1994).
[CrossRef]

Panariello, G.

G. Breglio, O. M. Bucci, A. Cutolo, R. Massa, G. Panariello, “Electro-optical sensing of GHz electromagnetic fields,” in Proceedings of International Conference on Electromagnetics in Advanced Applications 1997 (Politecnico di Torino, Torino, Italy, 1997), pp. 499–502.

Pascazio, V.

T. Isernia, V. Pascazio, R. Pierri, “A nonlinear estimation method in tomographic imaging,” IEEE Trans. Geosci. Remote Sens. 35, 910–923 (1997).
[CrossRef]

Paulsen, K. D.

P. M. Meaney, K. D. Paulsen, J. T. Chang, “Near-field microwave imaging of biologically-based materials using a monopole transceiver system,” IEEE Trans. Microwave Theory Tech. 46, 31–45 (1998).
[CrossRef]

Pierri, R.

T. Isernia, V. Pascazio, R. Pierri, “A nonlinear estimation method in tomographic imaging,” IEEE Trans. Geosci. Remote Sens. 35, 910–923 (1997).
[CrossRef]

R. Pierri, A. Brancaccio, “Imaging of a rotationally symmetric dielectric cylinder by a quadratic approach,” J. Opt. Soc. Am. A 14, 2777–2785 (1997).
[CrossRef]

T. Isernia, G. Leone, R. Pierri, F. Soldovieri, “On the local minima problem in phase reconstruction algorithms,” Radio Sci. 31, 1887–1899 (1996).
[CrossRef]

Porter, R. P.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products, A. Jeffrey, ed. (Academic, London, 1997).

Soldovieri, F.

T. Isernia, G. Leone, R. Pierri, F. Soldovieri, “On the local minima problem in phase reconstruction algorithms,” Radio Sci. 31, 1887–1899 (1996).
[CrossRef]

Spies, B. P.

T. M. Habashy, R. W. Groom, B. P. Spies, “Beyond the Born and Rytov approximations: a nonlinear approach to electromagnetic scattering,” J. Geophys. Res. 98, 1759–1775 (1993).
[CrossRef]

Stegun, I.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Wang, Y. M.

W. C. Chew, Y. M. Wang, G. Otto, D. Lesselier, J. Ch. Bolomey, “On the inverse source method of solving inverse scattering problems,” Inverse Probl. 10, 547–553 (1994).
[CrossRef]

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

Adv. Electron. Electron Phys. (1)

M. Bertero, “Linear inverse and ill-posed problems,” Adv. Electron. Electron Phys. 75, 1–120 (1989).
[CrossRef]

Electron. Commun. Eng. J. (1)

D. J. Daniels, “Surface penetrating radar,” Electron. Commun. Eng. J. 8, 165–182 (1996).
[CrossRef]

IEEE Microwave Guid. Wave Lett. (1)

W. C. Chew, J. H. Lin, “A frequency-hopping approach for microwave imaging of large inhomogeneous bodies,” IEEE Microwave Guid. Wave Lett. 5, 439–441 (1995).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

W. Liang, “A probe for making near-field measurement with minimal disturbance: the optically modulated scatterer,” IEEE Trans. Antennas Propag. 45, 772–779 (1997).
[CrossRef]

O. M. Bucci, G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antennas Propag. 37, 918–926 (1989).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

T. Isernia, V. Pascazio, R. Pierri, “A nonlinear estimation method in tomographic imaging,” IEEE Trans. Geosci. Remote Sens. 35, 910–923 (1997).
[CrossRef]

IEEE Trans. Med. Imaging (1)

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

IEEE Trans. Microwave Theory Tech. (2)

J. C. Bolomey, “Recent European developments in active microwave imaging for industrial, scientific, and medical applications,” IEEE Trans. Microwave Theory Tech. 37, 2109–2117 (1991).
[CrossRef]

P. M. Meaney, K. D. Paulsen, J. T. Chang, “Near-field microwave imaging of biologically-based materials using a monopole transceiver system,” IEEE Trans. Microwave Theory Tech. 46, 31–45 (1998).
[CrossRef]

Inverse Probl. (1)

W. C. Chew, Y. M. Wang, G. Otto, D. Lesselier, J. Ch. Bolomey, “On the inverse source method of solving inverse scattering problems,” Inverse Probl. 10, 547–553 (1994).
[CrossRef]

J. Appl. Geophys. (1)

F. N. Kong, T. L. By, “Performance of a GPR system which uses step frequency signals,” J. Appl. Geophys. 33, 15–26 (1995).
[CrossRef]

J. Geophys. Res. (1)

T. M. Habashy, R. W. Groom, B. P. Spies, “Beyond the Born and Rytov approximations: a nonlinear approach to electromagnetic scattering,” J. Geophys. Res. 98, 1759–1775 (1993).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Radio Sci. (2)

O. M. Bucci, T. Isernia, “Electromagnetic inverse scattering: retrievable information and measurement strategies,” Radio Sci. 32, 2123–2138 (1997).
[CrossRef]

T. Isernia, G. Leone, R. Pierri, F. Soldovieri, “On the local minima problem in phase reconstruction algorithms,” Radio Sci. 31, 1887–1899 (1996).
[CrossRef]

Other (4)

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products, A. Jeffrey, ed. (Academic, London, 1997).

This is equivalent to assuming that the actual field in a given point depends mainly on the incident field in that point, or, alternatively, that the value of the incident field in a given point affects only values of E in the points nearby.

G. Breglio, O. M. Bucci, A. Cutolo, R. Massa, G. Panariello, “Electro-optical sensing of GHz electromagnetic fields,” in Proceedings of International Conference on Electromagnetics in Advanced Applications 1997 (Politecnico di Torino, Torino, Italy, 1997), pp. 499–502.

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 (9)

Fig. 1
Fig. 1

Geometry of the problem.

Fig. 2
Fig. 2

Behavior of normalized singular values (in decibels), for different values of the dimension of the source RS, when the observation domain is placed at an infinite distance from the scatterer. The variable in the abscissa is the normalized index n/βRS.

Fig. 3
Fig. 3

Behavior of normalized singular values (the positive part of the spectrum is reported) as a function of the index n, where the distance d/λ between the probes and the object is taken as a parameter. The radius of the equivalent source is 5λ, where λ is the wavelength in the background medium.

Fig. 4
Fig. 4

Universal curve for the scaled close-proximity gain for different values of Ro. The variable to on the abscissa, defined in Eq. (10), depends both on the extension Ro of the observation domain and on the order n of the singular value. The scaling factor is (2/βRo)-1/3.

Fig. 5
Fig. 5

Universal curve for the far-zone scaled singular values for different values of RS. The variable tS on the abscissa, defined in Eq. (11), depends both on the extension RS of the scatterer and on the order n of the singular value. The scaling factor is (2/βRS)1/3.

Fig. 6
Fig. 6

Behavior of normalized singular values for the case of a square source domain of side 5λ, as a function of the index n, with the distance d/λ between the probes and the object being taken as a parameter. The whole spectrum is reported.

Fig. 7
Fig. 7

Behavior of the universal function K(tS) for different values of RS. Given RS and the maximum value of n, such a universal curve permits evaluation of the effective penetration depth.

Fig. 8
Fig. 8

Real part of the reference and reconstructed profiles: (a) reference, (b) intermediate configuration, (c) intermediate configuration with 3% measurement error, (d) full–near configuration with 3% measurement error, (e) full–near configuration with 5% measurement error, (f) full–near configuration with the primary sources and data corrupted by a 1.75% additive noise.

Fig. 9
Fig. 9

Case of a dielectric profile that presents fast spatial oscillations in the central part of the region under test: (a) real part of the reference profile, (b) real part of the reconstructed profile obtained in the full–near configuration.  

Equations (24)

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

ESν(r)=jωbDEν(r)χ(r)ge(r, r)dr=Ae(χEν),rD,
Eν(r)=Eincν(r)+jωbDEν(r)χ(r)gi(r, r)dr=Eincν(r)+Ai(χEν),rD,
χ(r)=(r)/b-1,
Ae[u]=nσnu(r, θ), un(r, θ)νn(θ),
Ae[u]=-j4 β2n=-+Hn2(βRo)02π0RSJn(βr)×exp[ jn(θ-θ)]u(r, θ)rdrdθ,
un(r, θ)=CnJn(βr)exp( jnθ),
νn(θ)=12π exp( jnθ).
σn2=πβ222|Hn2(βRo)|20RS[Jn(βr)]2rdr.
σ^n2=σn2σ02=Hn2(βRo)H02(βRo)2 [Jn(βRS)]2-Jn-1(βRS)Jn+1(βRS)[J0(βRS)]2+[J1(βRS)]2,
to=2βRo1/3(n-βRo),
tS=2βRS1/3(n-βRS)
ES(r^S, θo)n=-NN/2NN/2S(r^S, n)exp(jnθo),
ES(r^S, r^o)=ES(θS, θo)=m=-NN/2NN/2 n=-NN/2NN/2×S(m, n)exp( jn θo)exp( jm θS).
S(m, n)=σ^m(Ro)σ^n(Rt)CmCn0RSrdrJn(βr)Jm(βr)×02πdθχ(r, θ)exp[-j(m+n)θ],
|Cn|=12π 0RSJn2(βr)rdr-1/2,
|S(m, n)|=σ^m(Ro)σ^n(Rt)×0RSJ^n(βr)J^m(βr)Xm+n(r)rdr,
Xm+n(r)=12π 02πχ(r, θ)exp[-j(n+m)θ]dθ
βΔnn=βRS21/3K(tS),
Jeq=jωχ(I-Aiχ)-1Ei,
errχ=χrec-χrefχref,
RSΔnmJn(βRS)Jm(βRS)=0RSJn(βr)Jm(βr)rdr.
ΔnmRSJn(βr)RSJn(βRS) Jm(βr)RSJm(βRS),
Δnn=Jn(βr)2RSJn2(βRS)=RS2 Jn2(βRS)-Jn-1(βRS)Jn+1(βRS)Jn2(βRS).
βΔnn=βRS21/3K(tS),

Metrics