Abstract

A recently developed inverse scattering method based on the distorted-wave Born approximation (DWBA) that applies to objects embedded in known background media [Inverse Probl. 19, 855 (2003); 20, 1307 (2004) ] is implemented for the special case of circularly symmetric scatterers embedded in circularly symmetric backgrounds. The newly developed scheme is applied in a computer-simulation study of optical diffraction tomography (ODT), and the results are compared and contrasted with reconstructions obtained using the filtered backpropagation algorithm (FBP algorithm). Unlike the DWBA-based inversion algorithm, the FBP algorithm does not take into account multiple scattering within the known background, and it is found that the newly implemented scheme yields reconstructions much superior to those obtained using the FBP algorithm. The research reported applies to a number of important applications that include ultrasound nondestructive evaluation testing of cylinders for defects as well as ODT.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. J. Devaney and M. Dennison, "Inverse scattering in inhomogeneous background media," Inverse Probl. 19, 855-870 (2003).
    [CrossRef]
  2. M. Dennison and A. J. Devaney, "Inverse scattering in inhomogeneous background media II. Multi-frequency case and SVD formulation," Inverse Probl. 20, 1307-1324 (2004).
    [CrossRef]
  3. P. Guo and A. J. Devaney, "A comparison of reconstruction algorithms for optical diffraction tomography," J. Opt. Soc. Am. A 22, 2338-2347 (2005).
    [CrossRef]
  4. M. Maleki, A. J. Devaney, and A. Schatzberg, "Tomographic reconstruction from optical scattered intensities," J. Opt. Soc. Am. A 9, 1356 (1992).
    [CrossRef]
  5. T. C. Wedberg and J. J. Stamnes, "Experimental examination of the quantitative imaging properties of optical diffraction tomography," J. Opt. Soc. Am. A 12, 493-500 (1995).
    [CrossRef]
  6. T. C. Wedberg and W. C. Wedberg, "Tomographic reconstruction of the cross-sectional complex refractive index of semitransparent, birefringent fibers," J. Microsc. 177, 53-67 (1995).
    [CrossRef]
  7. M. Maleki and A. J. Devaney, "Phase retrieval and intensity-only reconstruction algorithm for optical diffraction tomography," J. Opt. Soc. Am. A 10, 1086-1092 (1993).
    [CrossRef]
  8. A. J. Devaney, "A filtered backpropagation algorithm for diffraction tomography," Ultrason. Imaging 4, 336-350 (1982).
    [CrossRef] [PubMed]
  9. W. Tabbara, B. Duchene, Ch. Pichot, D. Lesselier, L. Chommeloux, and N. Joachimowicz, "Diffraction tomography: contribution to the analysis of some applications in microwaves and ultrasonics," Inverse Probl. 4, 305-331 (1988).
    [CrossRef]
  10. http://www-eng.llnl.gov/pdfs/ndc-7.pdf.
  11. J. H. Taylor, Scattering Theory (Wiley, 1972).
  12. A. Sommerfeld, Partial Differential Equations (Academic, 1967), p. 189.
  13. C. K. Avinash and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1988).
  14. A. J. Devaney, "A computer simulation study of diffraction tomography," IEEE Trans. Biomed. Eng. 30, 377-386 (1983).
    [CrossRef] [PubMed]
  15. N. Joachimowicz, C. Pichot, and J. P. Hugonin, "Inverse scattering: an iterative numerical method for electromagnetic imaging," IEEE Trans. Antennas Propag. 39, 1742-1753 (1991).
    [CrossRef]
  16. A. N. Tikhonov, A. V. Concharsky, V. V. Stepanov, and A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems (Kluwer Academic, 1995).
  17. P. C. Hansen, "Analysis of discrete ill-posed problems by means of the 1-curve," SIAM Rev. 34, 561-580 (1992).
    [CrossRef]
  18. P. Guo and A. J. Devaney, "Digital microscopy using phase-shifting digital holography with two reference waves," Opt. Lett. 29, 857-859 (2004).
    [CrossRef] [PubMed]

2005 (1)

2004 (2)

P. Guo and A. J. Devaney, "Digital microscopy using phase-shifting digital holography with two reference waves," Opt. Lett. 29, 857-859 (2004).
[CrossRef] [PubMed]

M. Dennison and A. J. Devaney, "Inverse scattering in inhomogeneous background media II. Multi-frequency case and SVD formulation," Inverse Probl. 20, 1307-1324 (2004).
[CrossRef]

2003 (1)

A. J. Devaney and M. Dennison, "Inverse scattering in inhomogeneous background media," Inverse Probl. 19, 855-870 (2003).
[CrossRef]

1995 (3)

T. C. Wedberg and W. C. Wedberg, "Tomographic reconstruction of the cross-sectional complex refractive index of semitransparent, birefringent fibers," J. Microsc. 177, 53-67 (1995).
[CrossRef]

T. C. Wedberg and J. J. Stamnes, "Experimental examination of the quantitative imaging properties of optical diffraction tomography," J. Opt. Soc. Am. A 12, 493-500 (1995).
[CrossRef]

A. N. Tikhonov, A. V. Concharsky, V. V. Stepanov, and A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems (Kluwer Academic, 1995).

1993 (1)

1992 (2)

M. Maleki, A. J. Devaney, and A. Schatzberg, "Tomographic reconstruction from optical scattered intensities," J. Opt. Soc. Am. A 9, 1356 (1992).
[CrossRef]

P. C. Hansen, "Analysis of discrete ill-posed problems by means of the 1-curve," SIAM Rev. 34, 561-580 (1992).
[CrossRef]

1991 (1)

N. Joachimowicz, C. Pichot, and J. P. Hugonin, "Inverse scattering: an iterative numerical method for electromagnetic imaging," IEEE Trans. Antennas Propag. 39, 1742-1753 (1991).
[CrossRef]

1988 (2)

W. Tabbara, B. Duchene, Ch. Pichot, D. Lesselier, L. Chommeloux, and N. Joachimowicz, "Diffraction tomography: contribution to the analysis of some applications in microwaves and ultrasonics," Inverse Probl. 4, 305-331 (1988).
[CrossRef]

C. K. Avinash and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1988).

1983 (1)

A. J. Devaney, "A computer simulation study of diffraction tomography," IEEE Trans. Biomed. Eng. 30, 377-386 (1983).
[CrossRef] [PubMed]

1982 (1)

A. J. Devaney, "A filtered backpropagation algorithm for diffraction tomography," Ultrason. Imaging 4, 336-350 (1982).
[CrossRef] [PubMed]

1972 (1)

J. H. Taylor, Scattering Theory (Wiley, 1972).

1967 (1)

A. Sommerfeld, Partial Differential Equations (Academic, 1967), p. 189.

Avinash, C. K.

C. K. Avinash and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1988).

Chommeloux, L.

W. Tabbara, B. Duchene, Ch. Pichot, D. Lesselier, L. Chommeloux, and N. Joachimowicz, "Diffraction tomography: contribution to the analysis of some applications in microwaves and ultrasonics," Inverse Probl. 4, 305-331 (1988).
[CrossRef]

Concharsky, A. V.

A. N. Tikhonov, A. V. Concharsky, V. V. Stepanov, and A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems (Kluwer Academic, 1995).

Dennison, M.

M. Dennison and A. J. Devaney, "Inverse scattering in inhomogeneous background media II. Multi-frequency case and SVD formulation," Inverse Probl. 20, 1307-1324 (2004).
[CrossRef]

A. J. Devaney and M. Dennison, "Inverse scattering in inhomogeneous background media," Inverse Probl. 19, 855-870 (2003).
[CrossRef]

Devaney, A. J.

P. Guo and A. J. Devaney, "A comparison of reconstruction algorithms for optical diffraction tomography," J. Opt. Soc. Am. A 22, 2338-2347 (2005).
[CrossRef]

P. Guo and A. J. Devaney, "Digital microscopy using phase-shifting digital holography with two reference waves," Opt. Lett. 29, 857-859 (2004).
[CrossRef] [PubMed]

M. Dennison and A. J. Devaney, "Inverse scattering in inhomogeneous background media II. Multi-frequency case and SVD formulation," Inverse Probl. 20, 1307-1324 (2004).
[CrossRef]

A. J. Devaney and M. Dennison, "Inverse scattering in inhomogeneous background media," Inverse Probl. 19, 855-870 (2003).
[CrossRef]

M. Maleki and A. J. Devaney, "Phase retrieval and intensity-only reconstruction algorithm for optical diffraction tomography," J. Opt. Soc. Am. A 10, 1086-1092 (1993).
[CrossRef]

M. Maleki, A. J. Devaney, and A. Schatzberg, "Tomographic reconstruction from optical scattered intensities," J. Opt. Soc. Am. A 9, 1356 (1992).
[CrossRef]

A. J. Devaney, "A computer simulation study of diffraction tomography," IEEE Trans. Biomed. Eng. 30, 377-386 (1983).
[CrossRef] [PubMed]

A. J. Devaney, "A filtered backpropagation algorithm for diffraction tomography," Ultrason. Imaging 4, 336-350 (1982).
[CrossRef] [PubMed]

Duchene, B.

W. Tabbara, B. Duchene, Ch. Pichot, D. Lesselier, L. Chommeloux, and N. Joachimowicz, "Diffraction tomography: contribution to the analysis of some applications in microwaves and ultrasonics," Inverse Probl. 4, 305-331 (1988).
[CrossRef]

Guo, P.

Hansen, P. C.

P. C. Hansen, "Analysis of discrete ill-posed problems by means of the 1-curve," SIAM Rev. 34, 561-580 (1992).
[CrossRef]

Hugonin, J. P.

N. Joachimowicz, C. Pichot, and J. P. Hugonin, "Inverse scattering: an iterative numerical method for electromagnetic imaging," IEEE Trans. Antennas Propag. 39, 1742-1753 (1991).
[CrossRef]

Joachimowicz, N.

N. Joachimowicz, C. Pichot, and J. P. Hugonin, "Inverse scattering: an iterative numerical method for electromagnetic imaging," IEEE Trans. Antennas Propag. 39, 1742-1753 (1991).
[CrossRef]

W. Tabbara, B. Duchene, Ch. Pichot, D. Lesselier, L. Chommeloux, and N. Joachimowicz, "Diffraction tomography: contribution to the analysis of some applications in microwaves and ultrasonics," Inverse Probl. 4, 305-331 (1988).
[CrossRef]

Lesselier, D.

W. Tabbara, B. Duchene, Ch. Pichot, D. Lesselier, L. Chommeloux, and N. Joachimowicz, "Diffraction tomography: contribution to the analysis of some applications in microwaves and ultrasonics," Inverse Probl. 4, 305-331 (1988).
[CrossRef]

Maleki, M.

Pichot, C.

N. Joachimowicz, C. Pichot, and J. P. Hugonin, "Inverse scattering: an iterative numerical method for electromagnetic imaging," IEEE Trans. Antennas Propag. 39, 1742-1753 (1991).
[CrossRef]

Pichot, Ch.

W. Tabbara, B. Duchene, Ch. Pichot, D. Lesselier, L. Chommeloux, and N. Joachimowicz, "Diffraction tomography: contribution to the analysis of some applications in microwaves and ultrasonics," Inverse Probl. 4, 305-331 (1988).
[CrossRef]

Schatzberg, A.

Slaney, M.

C. K. Avinash and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1988).

Sommerfeld, A.

A. Sommerfeld, Partial Differential Equations (Academic, 1967), p. 189.

Stamnes, J. J.

Stepanov, V. V.

A. N. Tikhonov, A. V. Concharsky, V. V. Stepanov, and A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems (Kluwer Academic, 1995).

Tabbara, W.

W. Tabbara, B. Duchene, Ch. Pichot, D. Lesselier, L. Chommeloux, and N. Joachimowicz, "Diffraction tomography: contribution to the analysis of some applications in microwaves and ultrasonics," Inverse Probl. 4, 305-331 (1988).
[CrossRef]

Taylor, J. H.

J. H. Taylor, Scattering Theory (Wiley, 1972).

Tikhonov, A. N.

A. N. Tikhonov, A. V. Concharsky, V. V. Stepanov, and A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems (Kluwer Academic, 1995).

Wedberg, T. C.

T. C. Wedberg and J. J. Stamnes, "Experimental examination of the quantitative imaging properties of optical diffraction tomography," J. Opt. Soc. Am. A 12, 493-500 (1995).
[CrossRef]

T. C. Wedberg and W. C. Wedberg, "Tomographic reconstruction of the cross-sectional complex refractive index of semitransparent, birefringent fibers," J. Microsc. 177, 53-67 (1995).
[CrossRef]

Wedberg, W. C.

T. C. Wedberg and W. C. Wedberg, "Tomographic reconstruction of the cross-sectional complex refractive index of semitransparent, birefringent fibers," J. Microsc. 177, 53-67 (1995).
[CrossRef]

Yagola, A. G.

A. N. Tikhonov, A. V. Concharsky, V. V. Stepanov, and A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems (Kluwer Academic, 1995).

IEEE Trans. Antennas Propag. (1)

N. Joachimowicz, C. Pichot, and J. P. Hugonin, "Inverse scattering: an iterative numerical method for electromagnetic imaging," IEEE Trans. Antennas Propag. 39, 1742-1753 (1991).
[CrossRef]

IEEE Trans. Biomed. Eng. (1)

A. J. Devaney, "A computer simulation study of diffraction tomography," IEEE Trans. Biomed. Eng. 30, 377-386 (1983).
[CrossRef] [PubMed]

Inverse Probl. (3)

W. Tabbara, B. Duchene, Ch. Pichot, D. Lesselier, L. Chommeloux, and N. Joachimowicz, "Diffraction tomography: contribution to the analysis of some applications in microwaves and ultrasonics," Inverse Probl. 4, 305-331 (1988).
[CrossRef]

A. J. Devaney and M. Dennison, "Inverse scattering in inhomogeneous background media," Inverse Probl. 19, 855-870 (2003).
[CrossRef]

M. Dennison and A. J. Devaney, "Inverse scattering in inhomogeneous background media II. Multi-frequency case and SVD formulation," Inverse Probl. 20, 1307-1324 (2004).
[CrossRef]

J. Microsc. (1)

T. C. Wedberg and W. C. Wedberg, "Tomographic reconstruction of the cross-sectional complex refractive index of semitransparent, birefringent fibers," J. Microsc. 177, 53-67 (1995).
[CrossRef]

J. Opt. Soc. Am. A (4)

Opt. Lett. (1)

SIAM Rev. (1)

P. C. Hansen, "Analysis of discrete ill-posed problems by means of the 1-curve," SIAM Rev. 34, 561-580 (1992).
[CrossRef]

Ultrason. Imaging (1)

A. J. Devaney, "A filtered backpropagation algorithm for diffraction tomography," Ultrason. Imaging 4, 336-350 (1982).
[CrossRef] [PubMed]

Other (5)

A. N. Tikhonov, A. V. Concharsky, V. V. Stepanov, and A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems (Kluwer Academic, 1995).

http://www-eng.llnl.gov/pdfs/ndc-7.pdf.

J. H. Taylor, Scattering Theory (Wiley, 1972).

A. Sommerfeld, Partial Differential Equations (Academic, 1967), p. 189.

C. K. Avinash and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1988).

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 (7)

Fig. 1
Fig. 1

Simulation geometry. (a) A radially symmetric scattering object embedded in free space and illuminated with a plane wave. (b) The total field generated in the scattering experiment is measured over a CCD array located at the distance l 0 from the center of the object. (c) A radial cut through the index distribution of the object.

Fig. 2
Fig. 2

(Color online) Comparison of scattered field data generated by different scattering models for an object comprising two concentric cylinders: (a) real part, (b) imaginary part, (c) magnitude. The Born scattered field data were generated using the core index n c for the background medium.

Fig. 3
Fig. 3

(Color online) Reconstruction results for the piecewise-constant object from the exact scattered field data shown in Fig. 2. The top figure shows the reconstruction generated using the DWBA algorithm, and the bottom uses the FBP algorithm with k b as the background wavenumber.

Fig. 4
Fig. 4

(Color online) Reconstruction results for the piecewise-constant object from the DWBA scattered field data shown in Fig. 2. The top figure shows the reconstruction generated using the DWBA algorithm and the bottom uses the FBP algorithm with k b as the background wavenumber.

Fig. 5
Fig. 5

(Color online) Comparison of scattered field data generated by different scattering models for an object having a sinusoidally varying perturbation with respect to its cylindrical background: (a) real part, (b) imaginary part, (c) magnitude. The Born scattered field data were generated using the core index n c for the background medium.

Fig. 6
Fig. 6

(Color online) Reconstruction results for the object having a sinusoidally varying perturbation from the exact scattered field data shown in Fig. 5. The top figure shows the reconstruction generated using the DWBA algorithm, and the bottom uses the FBP algorithm with k b as the background wavenumber.

Fig. 7
Fig. 7

(Color online) (a) Reconstructions of the object having a sinusoidal varying perturbation from noisy DWBA data having signal-to-noise ratios ranging from S N = 5 dB to S N = 25 dB . (b) The rms error plotted versus the signal-to-noise ratio.

Tables (2)

Tables Icon

Table 1 rms Errors of Inversion for Constant Perturbation (Piecewise Object) by Four Reconstruction Strategies

Tables Icon

Table 2 rms Errors of Inversion for Sinusoidally Varying Perturbation (General Object) by Four Reconstruction Strategies

Equations (51)

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

ψ ( r , α ) = ψ ( in ) ( r , α ) + V d 3 r G b ( r , r ) O ( r ) ψ ( r , α ) .
ψ ( ρ , ϕ ) = ψ ( in ) ( ρ , ϕ ) + 0 a b ρ d ρ 0 2 π d ϕ G b ( ρ , ρ ; ϕ , ϕ ) O ( ρ ) ψ ( ρ , ϕ ) ,
[ 2 + k b 2 ( ρ ) ] ψ ( in ) ( ρ , ϕ ) = 0 ,
ψ ( in ) ( ρ , ϕ ) e i k 0 ρ cos ϕ + f b ( ϕ ) e i k 0 ρ ρ ,
[ 2 + k b 2 ( ρ ) ] G b ( ρ , ρ ; ϕ , ϕ ) = δ ( ρ ρ , ϕ ϕ ) ρ .
ψ ( ρ , ϕ ) ψ ( in ) ( ρ , ϕ ) + 0 a b ρ d ρ 0 2 π d ϕ G b ( ρ , ρ ; ϕ , ϕ ) O ( ρ ) ψ ( i n ) ( ρ , ϕ ) .
ψ ( ρ , ϕ ) = e i k 0 ρ cos ϕ + 0 a b ρ d ρ 0 2 π d ϕ G 0 ( ρ , ρ ; ϕ , ϕ ) O 0 ( ρ ) ψ ( ρ , ϕ ) ,
G 0 ( ρ , ρ ; ϕ , ϕ ) = i 2 H 0 ( k 0 r r ) ,
ψ ( ρ , ϕ ) e i k 0 ρ cos ϕ + 0 a b ρ d ρ 0 2 π d ϕ G 0 ( ρ , ρ ; ϕ , ϕ ) O 0 ( ρ ) e i k 0 ρ cos ϕ .
ψ ( s ) ( y n ) 0 a b ρ d ρ 0 2 π d ϕ G b ( ρ n , ρ ; ϕ n , ϕ ) O ( ρ ) ψ ( in ) ( ρ , ϕ ) ,
ψ ( s ) ( y n ) 0 a b ρ d ρ O ( ρ ) π n * ( ρ ) ,
π n ( ρ ) = 0 2 π d ϕ G b * ( ρ n , ρ ; ϕ n , ϕ ) ψ ( in ) * ( ρ , ϕ ) , ρ a b .
ψ ( s ) ( y n ) π n , O H V ,
f 1 , f 2 H V = 0 a b ρ d ρ f 1 * ( ρ ) f 2 ( ρ ) .
O ̂ ( ρ ) = n = 1 N C n π n ( ρ ) ,
ψ ( s ) ( y n ) = n = 1 N C n π n , π n H V .
ψ ( s ) ( y ) 0 a b ρ d ρ 0 2 π d ϕ G 0 ( ρ , ρ ; ϕ , ϕ ) O 0 ( ρ ) e i k 0 ρ cos ϕ ,
O ̂ 0 ( ρ ) = i k 0 π 0 k 0 K d K ψ ̃ ( s ) ( K ) e i γ l 0 J 0 ( 2 k 0 ( k 0 γ ) ) ,
ψ ̃ ( s ) ( K ) = d y ψ ( s ) ( y ) e i K y
k b ( ρ ) = { k b 0 ρ a b k 0 ρ > a b } .
E = [ j n r ( ρ j ) n ( ρ j ) 2 j n ( ρ j ) 2 ] 1 2 ,
δ k ( ρ ) = { δ k 0 ρ a 0 0 ρ > a 0 } ,
δ n ( ρ ) = { δ n sin ( 4 π ρ a 0 ) 0 ρ a 0 0 ρ > a 0 } .
S N = 20 log max ψ ( s ) ( y n ) σ n ,
C ̂ = argmin C D Π C 2 2 + λ L C 2 2 ,
C ̂ = ( Π T Π + λ L T L ) 1 Π T D .
ψ ( in ) ( ρ , ϕ ) = m = { α m J m ( k b ρ ) e i m ϕ ρ < a b [ i m J m ( k 0 ρ ) + β m H m ( k 0 ρ ) ] e i m ϕ ρ > a b } .
α m J m ( k b a b ) = i m J m ( k 0 a b ) + β m H m ( k 0 a b ) ,
α m d d ρ J m ( k b ρ ) ρ = a b = i m d d ρ J m ( k 0 ρ ) ρ = a b + β m d d ρ H m ( k 0 ρ ) ρ = a b ,
G b ( ρ , ρ , ϕ , ϕ ) = m = g m ( ρ , ρ ) e i m ( ϕ ϕ ) ,
g m ( ρ , ρ ) = { A m ( ρ ) J m ( k b ρ ) 0 ρ < ρ [ B m ( ρ ) J m ( k b ρ ) + C m ( ρ ) Y m ( k b ρ ) ] ρ < ρ a b D m ( ρ ) H m ( k 0 ρ ) ρ < a b < ρ } .
B m ( ρ ) J m ( k b a b ) + C m ( ρ ) Y m ( k b a b ) = D m ( ρ ) H m ( k 0 a b ) ,
B m ( ρ ) d J m ( k b ρ ) d ρ ρ = a b + C m ( ρ ) d Y m ( k b ρ ) d ρ ρ = a b = D m ( ρ ) d H m ( k b ρ ) d ρ ρ = a b ;
A m ( ρ ) J m ( k b ρ ) = B m ( ρ ) J m ( k b ρ ) + C m ( ρ ) Y m ( k b ρ ) ,
B m ( ρ ) d J m ( k b ρ ) d ρ ρ = ρ + C m ( ρ ) d Y m ( k b ρ ) d ρ ρ = ρ A m ( ρ ) d J m ( k b ρ ) d ρ ρ = ρ = 1 2 π ρ ρ = ρ ,
π n ( ρ ) = 0 2 π d ϕ G b * ( ρ n , ρ ; ϕ n , ϕ ) ψ ( in ) * ( ρ , ϕ ) = 2 π m = α m D m ( ρ ) J m ( k b ρ ) H m ( k 0 ρ n ) e i m ϕ n .
ψ ( ρ , ϕ ) = m = { [ i m J m ( k 0 ρ ) + a m H m ( k 0 ρ ) ] e i m ϕ ρ > a b [ b m J m ( k b ρ ) + c m H m ( k b ρ ) ] e i m ϕ a 0 < ρ < a b d m J m [ ( k b + δ k ) ρ ] e i m ϕ ρ < a 0 } .
i m J m ( k 0 a b ) + a m H m ( k 0 a b ) = b m J m ( k b a b ) + c m H m ( k b a b ) ,
i m d J m ( k 0 ρ ) H V ρ ρ = a b + a m d H m ( k 0 ρ ) d ρ ρ = a b = b m d J m ( k b ρ ) d ρ ρ = a b + c n d H m ( k b ρ ) d ρ ρ = a b ;
b m J m ( k b a 0 ) + c m H m ( k b a 0 ) = d m J m ( ( k b + δ k ) a 0 ) ,
b m d J m ( k b ρ ) d ρ ρ = a 0 + c m d H m ( k b ρ ) d ρ ρ = a 0 = d m d J m ( ( k b + δ k ) ρ ) d ρ ρ = a 0 .
ψ ( s ) ( ρ n , ϕ n ) = m = a m H m ( k 0 ρ n ) e i m ϕ n .
m = Ψ m ( ρ ) e i m ϕ = m = α m J m ( k b ρ ) e i m ϕ + 0 a b ρ d ρ 0 2 π d ϕ [ m = g m ( ρ , ρ ) e i m ( ϕ ϕ ) ] O ( ρ ) m = Ψ m ( ρ ) e i m ϕ ,
Ψ m ( ρ ) = α m J m ( k b ρ ) + 2 π 0 a b ρ d ρ g m ( ρ , ρ ) O ( ρ ) Ψ m ( ρ ) ;
0 a b d ρ [ δ ( ρ ρ ) 2 π ρ g m ( ρ , ρ ) O ( ρ ) ] Ψ m ( ρ ) = α m J m ( k b ρ ) .
δ ρ n = 1 N [ 1 δ ρ δ n , n 2 π n δ ρ g m ( n δ ρ , n δ ρ ) O ( n δ ρ ) ] Ψ m ( n δ ρ ) = α m J m ( k b n δ ρ ) .
K n , n ( m ) = δ ρ [ δ n , n n δ ρ g m ( n δ ρ , n δ ρ ) O ( n δ ρ ) ]
Ψ ( m ) = [ Ψ m ( δ ρ ) , , Ψ m ( n δ ρ ) ] T ,
χ ( m ) = α m [ J m ( k b δ ρ ) , , J m ( k b n δ ρ ) ] T ,
[ K ( m ) ] Ψ ( m ) = χ ( m )
Ψ ( s ) ( ρ n , ϕ n ) = 2 π m = H m ( k 0 ρ n ) e i m ϕ n 0 a b ρ d ρ O ( ρ ) D m ( ρ ) Ψ m ( ρ ) .

Metrics