Abstract

We present an optical tomographic reconstruction method to recover the complex refractive index distribution from boundary measurements based on intensity, which are the logarithm of intensity and normal derivative of intensity. The method, which is iterative, repeatedly implements the forward propagation equation for light amplitude, the Helmholtz equation, and computes appropriate sensitivity matrices for these measurements. The sensitivity matrices are computed by solving the forward propagation equation for light and its adjoint. The results of numerical experiments show that the data types ln(I) and In reconstructed, respectively, the imaginary and the real part of the object refractive index distribution. Moreover, the imaginary part of the refractive index reconstructed from In and the real part from ln(I) failed to show the object’s inhomogeneity. The value of the propagation constant, k, used in our simulations was 50, and this value resulted in smoothing of the reconstructed inhomogeneities. Thus we have shown that it is possible to reconstruct the complex refractive index distribution directly from the measured intensity without having to first find the light amplitude, as is done in most of the currently available reconstruction algorithms of diffraction tomography.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Soumekh and J. H. Choi, "Reconstruction in diffraction imaging," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 36, 93-100 (1989).
    [CrossRef] [PubMed]
  2. 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]
  3. S. Gutman and M. Klibanov, "Regularized quasi-Newton method for inverse scattering problems," Math. Comput. Modell. 18, 5-31 (1993).
    [CrossRef]
  4. R. E. Kleinmann and P. M. Van den Berg, "Modified gradient method for 2-d problem in tomography," J. Comput. Appl. Math. 42, 17-35 (1992).
    [CrossRef]
  5. A. C. Kak and M. Slaney, "Tomographic imaging with diffracting sources," in Principles of Computerized Tomographic Imaging (IEEE, 1988), pp. 211-217.
  6. F. Natterer and F. Wübbeling, "A propagation-backpropagation method for ultrasound tomography," Inverse Probl. 11, 1225-1232 (1995).
    [CrossRef]
  7. S. S. Cha and H. Sun, "Tomography for reconstructing continuous fields from ill-posed multidirectional interferometric data," Appl. Opt. 29, 251-258 (1990).
    [CrossRef] [PubMed]
  8. T. E. Gureyev, A. Roberts, and K. A. Nugent, "Phase retrieval with the transport of intensity equation: matrix solution with use of zernite polynomial," J. Opt. Soc. Am. A 12, 1932-1941 (1995).
    [CrossRef]
  9. K. Ichikawa, A. W. Lohmann, and M. Takeda, "Phase retrieval based on the irradiance transport equation and the Fourier transform method: experiments," Appl. Opt. 27, 3433-3436 (1988).
    [CrossRef] [PubMed]
  10. T. C. Wedberg and J. J. Stamnes, "Comparison of phase retrieval methods in optical diffraction tomography," Pure Appl. Opt. 4, 39-54 (1995).
    [CrossRef]
  11. A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, "Quantitative optical phase microscopy," Opt. Lett. 23, 817-819 (1998).
    [CrossRef]
  12. G. Gbur and E. Wolf, "Hybrid diffraction tomography without phase information," J. Opt. Soc. Am. A 19, 2194-2202 (2002).
    [CrossRef]
  13. M. H. Maleki and A. J. Devaney, "Phase retrieval and intensity-only reconstruction algorithms for optical diffraction tomography," J. Opt. Soc. Am. A 10, 1086-1092 (1993).
    [CrossRef]
  14. J. Cheng and S. Han, "Diffraction tomography reconstruction algorithms for quantitative imaging of phase objects," J. Opt. Soc. Am. A 18, 1460-1464 (2001).
    [CrossRef]
  15. M. H. Maleki, A. J. Devaney, and A. Schatzberg, "Tomographic reconstruction from optical scattered intensities," J. Opt. Soc. Am. A 9, 1356-1363 (1992).
    [CrossRef]
  16. M. R. Teague, "Deterministic phase retrieval: a Green's function solution," J. Opt. Soc. Am. 73, 1434-1441 (1983).
    [CrossRef]
  17. G. Vdovin, "Reconstrution of an object shape from the near-field intensity of a reflected paraxial beam," Appl. Opt. 36, 5508-5513 (1997).
    [CrossRef] [PubMed]
  18. N. Jayashree, G. Keshava Datta, and R. M. Vasu, "Optical tomographic microscope for quantitaive imaging of phase objects," Appl. Opt. 39, 277-283 (2000).
    [CrossRef]
  19. K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, "Quantitative phase imaging using hard x-rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
    [CrossRef] [PubMed]
  20. A. Semichaevsky and M. Testorf, "Phase-space interpretation of deterministic phase retrieval," J. Opt. Soc. Am. A 21, 2173-2179 (2004).
    [CrossRef]
  21. A. H. Hielscher, A. D. Klose, and K. M. Hansen, "Gradient-based iterative image reconstruction scheme for time-resolved optical tomography," IEEE Trans. Med. Imaging 18, 262-271 (1999).
    [CrossRef] [PubMed]
  22. S. R. Arridge and M. Schweiger, "A gradient based optimization scheme for optical tomography," Opt. Express 6, 213-226 (1998).
    [CrossRef]
  23. S. R. Arridge, "Topical review: optical tomography in medical imaging," Inverse Probl. 15, R41-R93 (1999).
    [CrossRef]
  24. S. R. Arridge, "Photon-measurement density functions. Part 1: analytical forms," Appl. Opt. 34, 7395-7409 (1995).
    [CrossRef] [PubMed]
  25. S. R. Arridge, "Photon-measurement density functions. Part 2: finite-element-method calculations," Appl. Opt. 34, 8026-8037 (1995).
    [CrossRef] [PubMed]
  26. B. Kanmani and R. M. Vasu, "Diffuse optical tomography through solving a system of quadratic equation: theory and simulation," Phys. Med. Biol. 51, 981-998 (2006).
    [CrossRef] [PubMed]
  27. J. Shen and L. L. Wang, "Spectral approximation of the Helmholtz equation with high wave numbers," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 43, 623-644 (2005).
    [CrossRef]
  28. S. Chandler-Wilde and S. Langdon, "A wave number independent boundary element method for an acoustic scattering problem," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 43, 2450-2477 (2006).
    [CrossRef]
  29. S. Kim, C. S. Shin, and J. B. Keller, "High-frequency asymptotics for the numerical solution of Helmholtz equation," Appl. Math. Lett. 18, 797-804 (2005).
    [CrossRef]
  30. F. Natterer, "Marching schemes for inverse Helmholtz and Maxwell problems," http://arachne.uni-muenster.de:8000/num/Preprints.
  31. W. Dahmen, "Wavelet and multiscale methods for operator equations," Acta Numerica 6, 55-228 (1997).
    [CrossRef]

2006 (2)

B. Kanmani and R. M. Vasu, "Diffuse optical tomography through solving a system of quadratic equation: theory and simulation," Phys. Med. Biol. 51, 981-998 (2006).
[CrossRef] [PubMed]

S. Chandler-Wilde and S. Langdon, "A wave number independent boundary element method for an acoustic scattering problem," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 43, 2450-2477 (2006).
[CrossRef]

2005 (2)

S. Kim, C. S. Shin, and J. B. Keller, "High-frequency asymptotics for the numerical solution of Helmholtz equation," Appl. Math. Lett. 18, 797-804 (2005).
[CrossRef]

J. Shen and L. L. Wang, "Spectral approximation of the Helmholtz equation with high wave numbers," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 43, 623-644 (2005).
[CrossRef]

2004 (1)

2002 (1)

2001 (1)

2000 (1)

1999 (2)

A. H. Hielscher, A. D. Klose, and K. M. Hansen, "Gradient-based iterative image reconstruction scheme for time-resolved optical tomography," IEEE Trans. Med. Imaging 18, 262-271 (1999).
[CrossRef] [PubMed]

S. R. Arridge, "Topical review: optical tomography in medical imaging," Inverse Probl. 15, R41-R93 (1999).
[CrossRef]

1998 (2)

1997 (2)

1996 (1)

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, "Quantitative phase imaging using hard x-rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

1995 (6)

1993 (2)

S. Gutman and M. Klibanov, "Regularized quasi-Newton method for inverse scattering problems," Math. Comput. Modell. 18, 5-31 (1993).
[CrossRef]

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

1992 (2)

M. H. Maleki, A. J. Devaney, and A. Schatzberg, "Tomographic reconstruction from optical scattered intensities," J. Opt. Soc. Am. A 9, 1356-1363 (1992).
[CrossRef]

R. E. Kleinmann and P. M. Van den Berg, "Modified gradient method for 2-d problem in tomography," J. Comput. Appl. Math. 42, 17-35 (1992).
[CrossRef]

1990 (1)

1989 (1)

M. Soumekh and J. H. Choi, "Reconstruction in diffraction imaging," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 36, 93-100 (1989).
[CrossRef] [PubMed]

1988 (1)

1983 (1)

Arridge, S. R.

Barnea, Z.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, "Quantitative phase imaging using hard x-rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

Barty, A.

Cha, S. S.

Chandler-Wilde, S.

S. Chandler-Wilde and S. Langdon, "A wave number independent boundary element method for an acoustic scattering problem," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 43, 2450-2477 (2006).
[CrossRef]

Cheng, J.

Choi, J. H.

M. Soumekh and J. H. Choi, "Reconstruction in diffraction imaging," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 36, 93-100 (1989).
[CrossRef] [PubMed]

Cookson, D. F.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, "Quantitative phase imaging using hard x-rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

Dahmen, W.

W. Dahmen, "Wavelet and multiscale methods for operator equations," Acta Numerica 6, 55-228 (1997).
[CrossRef]

Devaney, A. J.

Gbur, G.

Gureyev, T. E.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, "Quantitative phase imaging using hard x-rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

T. E. Gureyev, A. Roberts, and K. A. Nugent, "Phase retrieval with the transport of intensity equation: matrix solution with use of zernite polynomial," J. Opt. Soc. Am. A 12, 1932-1941 (1995).
[CrossRef]

Gutman, S.

S. Gutman and M. Klibanov, "Regularized quasi-Newton method for inverse scattering problems," Math. Comput. Modell. 18, 5-31 (1993).
[CrossRef]

Han, S.

Hansen, K. M.

A. H. Hielscher, A. D. Klose, and K. M. Hansen, "Gradient-based iterative image reconstruction scheme for time-resolved optical tomography," IEEE Trans. Med. Imaging 18, 262-271 (1999).
[CrossRef] [PubMed]

Hielscher, A. H.

A. H. Hielscher, A. D. Klose, and K. M. Hansen, "Gradient-based iterative image reconstruction scheme for time-resolved optical tomography," IEEE Trans. Med. Imaging 18, 262-271 (1999).
[CrossRef] [PubMed]

Ichikawa, K.

Jayashree, N.

Kak, A. C.

A. C. Kak and M. Slaney, "Tomographic imaging with diffracting sources," in Principles of Computerized Tomographic Imaging (IEEE, 1988), pp. 211-217.

Kanmani, B.

B. Kanmani and R. M. Vasu, "Diffuse optical tomography through solving a system of quadratic equation: theory and simulation," Phys. Med. Biol. 51, 981-998 (2006).
[CrossRef] [PubMed]

Keller, J. B.

S. Kim, C. S. Shin, and J. B. Keller, "High-frequency asymptotics for the numerical solution of Helmholtz equation," Appl. Math. Lett. 18, 797-804 (2005).
[CrossRef]

Keshava Datta, G.

Kim, S.

S. Kim, C. S. Shin, and J. B. Keller, "High-frequency asymptotics for the numerical solution of Helmholtz equation," Appl. Math. Lett. 18, 797-804 (2005).
[CrossRef]

Kleinmann, R. E.

R. E. Kleinmann and P. M. Van den Berg, "Modified gradient method for 2-d problem in tomography," J. Comput. Appl. Math. 42, 17-35 (1992).
[CrossRef]

Klibanov, M.

S. Gutman and M. Klibanov, "Regularized quasi-Newton method for inverse scattering problems," Math. Comput. Modell. 18, 5-31 (1993).
[CrossRef]

Klose, A. D.

A. H. Hielscher, A. D. Klose, and K. M. Hansen, "Gradient-based iterative image reconstruction scheme for time-resolved optical tomography," IEEE Trans. Med. Imaging 18, 262-271 (1999).
[CrossRef] [PubMed]

Langdon, S.

S. Chandler-Wilde and S. Langdon, "A wave number independent boundary element method for an acoustic scattering problem," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 43, 2450-2477 (2006).
[CrossRef]

Lohmann, A. W.

Maleki, M. H.

Natterer, F.

F. Natterer and F. Wübbeling, "A propagation-backpropagation method for ultrasound tomography," Inverse Probl. 11, 1225-1232 (1995).
[CrossRef]

F. Natterer, "Marching schemes for inverse Helmholtz and Maxwell problems," http://arachne.uni-muenster.de:8000/num/Preprints.

Nugent, K. A.

Paganin, D.

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, "Quantitative optical phase microscopy," Opt. Lett. 23, 817-819 (1998).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, "Quantitative phase imaging using hard x-rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

Roberts, A.

Schatzberg, A.

Schweiger, M.

Semichaevsky, A.

Shen, J.

J. Shen and L. L. Wang, "Spectral approximation of the Helmholtz equation with high wave numbers," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 43, 623-644 (2005).
[CrossRef]

Shin, C. S.

S. Kim, C. S. Shin, and J. B. Keller, "High-frequency asymptotics for the numerical solution of Helmholtz equation," Appl. Math. Lett. 18, 797-804 (2005).
[CrossRef]

Slaney, M.

A. C. Kak and M. Slaney, "Tomographic imaging with diffracting sources," in Principles of Computerized Tomographic Imaging (IEEE, 1988), pp. 211-217.

Soumekh, M.

M. Soumekh and J. H. Choi, "Reconstruction in diffraction imaging," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 36, 93-100 (1989).
[CrossRef] [PubMed]

Stamnes, J. J.

T. C. Wedberg and J. J. Stamnes, "Comparison of phase retrieval methods in optical diffraction tomography," Pure Appl. Opt. 4, 39-54 (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]

Sun, H.

Takeda, M.

Teague, M. R.

Testorf, M.

Van den Berg, P. M.

R. E. Kleinmann and P. M. Van den Berg, "Modified gradient method for 2-d problem in tomography," J. Comput. Appl. Math. 42, 17-35 (1992).
[CrossRef]

Vasu, R. M.

B. Kanmani and R. M. Vasu, "Diffuse optical tomography through solving a system of quadratic equation: theory and simulation," Phys. Med. Biol. 51, 981-998 (2006).
[CrossRef] [PubMed]

N. Jayashree, G. Keshava Datta, and R. M. Vasu, "Optical tomographic microscope for quantitaive imaging of phase objects," Appl. Opt. 39, 277-283 (2000).
[CrossRef]

Vdovin, G.

Wang, L. L.

J. Shen and L. L. Wang, "Spectral approximation of the Helmholtz equation with high wave numbers," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 43, 623-644 (2005).
[CrossRef]

Wedberg, T. C.

T. C. Wedberg and J. J. Stamnes, "Comparison of phase retrieval methods in optical diffraction tomography," Pure Appl. Opt. 4, 39-54 (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]

Wolf, E.

Wübbeling, F.

F. Natterer and F. Wübbeling, "A propagation-backpropagation method for ultrasound tomography," Inverse Probl. 11, 1225-1232 (1995).
[CrossRef]

Acta Numerica (1)

W. Dahmen, "Wavelet and multiscale methods for operator equations," Acta Numerica 6, 55-228 (1997).
[CrossRef]

Appl. Math. Lett. (1)

S. Kim, C. S. Shin, and J. B. Keller, "High-frequency asymptotics for the numerical solution of Helmholtz equation," Appl. Math. Lett. 18, 797-804 (2005).
[CrossRef]

Appl. Opt. (6)

IEEE Trans. Med. Imaging (1)

A. H. Hielscher, A. D. Klose, and K. M. Hansen, "Gradient-based iterative image reconstruction scheme for time-resolved optical tomography," IEEE Trans. Med. Imaging 18, 262-271 (1999).
[CrossRef] [PubMed]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

M. Soumekh and J. H. Choi, "Reconstruction in diffraction imaging," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 36, 93-100 (1989).
[CrossRef] [PubMed]

Inverse Probl. (2)

F. Natterer and F. Wübbeling, "A propagation-backpropagation method for ultrasound tomography," Inverse Probl. 11, 1225-1232 (1995).
[CrossRef]

S. R. Arridge, "Topical review: optical tomography in medical imaging," Inverse Probl. 15, R41-R93 (1999).
[CrossRef]

J. Comput. Appl. Math. (1)

R. E. Kleinmann and P. M. Van den Berg, "Modified gradient method for 2-d problem in tomography," J. Comput. Appl. Math. 42, 17-35 (1992).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (7)

Math. Comput. Modell. (1)

S. Gutman and M. Klibanov, "Regularized quasi-Newton method for inverse scattering problems," Math. Comput. Modell. 18, 5-31 (1993).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Med. Biol. (1)

B. Kanmani and R. M. Vasu, "Diffuse optical tomography through solving a system of quadratic equation: theory and simulation," Phys. Med. Biol. 51, 981-998 (2006).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, "Quantitative phase imaging using hard x-rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

Pure Appl. Opt. (1)

T. C. Wedberg and J. J. Stamnes, "Comparison of phase retrieval methods in optical diffraction tomography," Pure Appl. Opt. 4, 39-54 (1995).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. (2)

J. Shen and L. L. Wang, "Spectral approximation of the Helmholtz equation with high wave numbers," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 43, 623-644 (2005).
[CrossRef]

S. Chandler-Wilde and S. Langdon, "A wave number independent boundary element method for an acoustic scattering problem," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 43, 2450-2477 (2006).
[CrossRef]

Other (2)

F. Natterer, "Marching schemes for inverse Helmholtz and Maxwell problems," http://arachne.uni-muenster.de:8000/num/Preprints.

A. C. Kak and M. Slaney, "Tomographic imaging with diffracting sources," in Principles of Computerized Tomographic Imaging (IEEE, 1988), pp. 211-217.

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 (6)

Fig. 1
Fig. 1

MOBIIR algorithm: the inputs are the experimental measurement M e and the initial guess of the optical property f 0 . The algorithm has two iterative loops, the outer and the inner. In the outer loop, the perturbation equation is updated, and in the inner loop the perturbation equation is solved for the optical property distribution.

Fig. 2
Fig. 2

Data collection geometry: the object, which is circular, is enclosed in a square region bounded by L L L + . The source is a monochromatic plane wave incident on L , and the transmitted intensity is detected at L + . Data for different views are gathered by rotating the circular region, which contains the inhomogeneity in refractive index.

Fig. 3
Fig. 3

(a) Real part of the input object refractive index distribution. (b) Reconstructed image obtained from the normal derivative of the measured intensity at the boundary. (c) Cross-sectional plots through the center of the reconstruction in (a), curve (2) and the original object, curve (1).

Fig. 4
Fig. 4

(a) Imaginary part of the input object refractive index distribution. (b) Reconstructed image obtained from the logarithm of the measured intensity at the boundary. (c) Cross-sectional plots through the center of the reconstruction in (a), curve (2) and the original object, curve (1).

Fig. 5
Fig. 5

Cross section through the center of the inhomogeneity of the reconstructions from I n . Curve (a), for real part of the original object; curve (b), real part of the reconstructed refractive index; curve (c), imaginary part of the reconstructed refractive index.

Fig. 6
Fig. 6

Cross section through the center of the inhomogeneity of the reconstructions from ln ( I ) . Curve (a), imaginary part of the original object; curve (b), imaginary part of the reconstructed refractive index; curve (c), real part of the reconstructed refractive index.

Equations (64)

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

u ( r ) + k 2 u ( r ) = 0 .
u ( r ) + k 0 2 ( 1 f ( r ) ) u ( r ) = 0 ,
f ( r ) = 2 n δ ( r ) , k 0 = 2 π λ 0 .
θ = ( sin ϕ cos ϕ ) ,
u ( r ) = e i k 0 r θ ( 1 + v ( r ) ) .
v ( r ) + 2 i k 0 θ v ( r ) k 0 2 v ( r ) f ( r ) = k 0 2 f ( r ) .
v ( r ) = g θ on L , v ( r ) = 0 on L ,
v ( r ) + v ( r ) n = 0 on L + .
v δ ( r ) + 2 i k 0 θ v δ ( r ) k 0 2 v δ ( r ) f ( r ) = k 0 2 ( 1 + v ( r ) ) d ( r ) .
v δ ( r ) = 0 on L L , v δ ( r ) + v δ ( r ) n = 0 on L + .
Γ I δ = ln ( u u ¯ + u u ¯ δ + u δ u ¯ + u δ u ¯ δ ) ln ( u u ¯ ) = ln ( u u ¯ + u u ¯ δ + u δ u ¯ + u δ u ¯ δ u u ¯ ) ln ( 1 + u u ¯ δ + u δ u ¯ u u ¯ ) .
Γ I δ = u u ¯ δ + u δ u ¯ u u ¯ = ( u ¯ δ u ¯ ) + ( u δ u ) .
u δ = ( 1 + v + v δ ) e i k 0 r θ ( 1 + v ) e i k 0 r θ = v δ e i k 0 r θ .
v δ = u Γ I , 1 δ e i k 0 r θ .
v δ n = u Γ I , 1 δ e i k 0 r θ .
ψ ( r ) + 2 i k 0 θ ψ ( r ) k 0 2 f ¯ ( r ) ψ ( r ) = 0
ψ ( r ) n + ψ ( r ) ( 1 + 2 i k 0 θ n ) = q + on L + ,
ψ ( r ) = 0 on L L .
L + [ ψ ¯ n + ψ ¯ ( 1 2 i k 0 θ n ) ] v δ n d n 1 r = Ω [ k 2 ( 1 + v ( r ) ) d ( r ) ) ψ ¯ ( r ) d n r .
L + [ q ¯ + v δ ( r ) n ] d n 1 r = Ω k 0 2 ( 1 + v ( r ) ) d ( r ) ψ ¯ ( r ) d n r .
Γ I , 1 δ = 1 u v δ n e i k 0 r θ ,
Γ I , 1 δ ( m 0 ) = e i k 0 r m 0 u ( m 0 ) Ω k 0 2 ( 1 + v ( r ) ) d ( r ) G ¯ R ψ ( r , m 0 ) d n r .
v δ ( m 0 ) = Ω [ G v ( m 0 , r ) k 0 2 ( 1 + v ( r ) ) d ( r ) d n r ] .
Γ I , 1 δ ( m 0 ) = e i k 0 r m 0 u ( m 0 ) Ω k 0 2 ( 1 + v ( r ) ) d ( r ) G Γ I , 1 δ ( m 0 , r ) d n r .
G Γ I , 1 δ ( m 0 , r ) = G ¯ R ψ ( r , m 0 ) .
G ¯ Γ I , 2 δ ( m 0 , r ) = G R ψ ( r , m 0 ) .
Γ I ( m 0 , r ) d ( r ) = e i k 0 r m 0 u ( m 0 ) k 0 2 ( 1 + v ( r ) ) G Γ I , 1 δ ( m 0 , r ) + e i k 0 r m 0 u ¯ ( m 0 ) k 0 2 ( 1 + v ¯ ( r ) ) G ¯ Γ I , 2 δ ( m 0 , r ) .
Γ N δ = [ u u ¯ δ + u δ u ¯ + u δ u ¯ δ ] n .
Γ N δ = [ u u ¯ δ + u δ u ¯ ] n .
Γ N δ = u ¯ u δ n + u ¯ δ u n + u u ¯ δ n + u δ u ¯ n
= Γ N , 1 δ + Γ N , 2 δ + Γ N , 3 δ + Γ N , 4 δ .
Γ N , 1 δ = u ¯ [ v δ e i k 0 r θ ] n .
v δ n = Γ N , 1 δ v δ u ¯ [ e i k 0 r θ ] n u ¯ e i k 0 r θ
= Γ N , 1 δ + v δ n u ¯ [ e i k 0 r θ ] n u ¯ e i k 0 r θ .
[ e i k 0 r θ ] n = e i k 0 r θ i k 0 cos ϕ .
[ v δ n ] L + = Γ N , 1 δ u ¯ e i k 0 r θ [ 1 1 i k 0 cos ϕ ] .
Γ N , 1 δ ( m 0 ) = e i k 0 r m 0 u ¯ ( m 0 ) [ 1 i k 0 cos ϕ ] Ω k 0 2 ( 1 + v ( r ) ) d ( r ) G ¯ R ψ ( r , m 0 ) d n r .
v δ ( m 0 ) = Ω [ G v ( m 0 , r ) k 0 2 ( 1 + v ( r ) ) d ( r ) ] d n r .
Γ N , 1 δ ( m 0 ) = u ¯ ( m 0 ) e i k 0 r m 0 ( 1 i k 0 cos ϕ ) Ω G Γ N , 1 ( m 0 , r ) k 0 2 ( 1 + v ( r ) ) d ( r ) d n r .
G ¯ R ψ ( r , m 0 ) = G Γ N , 1 ( m 0 , r ) .
Γ N , 2 δ = u n u ¯ δ .
v δ = Γ ¯ N , 2 δ e i k 0 r θ u ¯ n .
Γ N , 2 δ ( m 0 ) = e i k 0 r m 0 u ( m 0 ) n Ω k 0 2 ( 1 + v ¯ ( r ) ) d ¯ ( r ) G R ψ ( r , m 0 ) d n r ,
Γ N , 2 δ ( m 0 ) = u ( m 0 ) n e i k 0 r m 0 Ω G ¯ Γ N , 2 ( m 0 , r ) k 0 2 ( 1 + v ¯ ( r ) ) d ¯ ( r ) d n r .
G R ψ ( r , m 0 ) = G ¯ Γ N , 2 ( m 0 , r ) .
Γ N , 3 δ ( m 0 ) = e i k 0 r m 0 u ( m 0 ) [ 1 + i k 0 cos ϕ ] Ω k 0 2 ( 1 + v ¯ ( r ) ) d ¯ ( r ) G R ψ ( r , m 0 ) d n r .
G R ψ ( r , m 0 ) = G ¯ Γ N , 3 ( m 0 , r ) .
Γ N , 4 δ ( m 0 ) = e i k 0 r m 0 u ¯ ( m 0 ) n Ω k 0 2 ( 1 + v ( r ) ) d ( r ) G ¯ R ψ ( r , m 0 ) d n r .
G ¯ R ψ ( r , m 0 ) = G Γ N , 4 ( m 0 , r ) .
Γ N ( m 0 , r ) d ( r ) = e i k 0 r m 0 u ¯ ( m 0 ) k 0 2 ( 1 i k 0 cos ϕ ) ( 1 + v ( r ) ) G Γ N , 1 ( m 0 , r ) e i k 0 r m 0 u ( m 0 ) n k 0 2 ( 1 + v ¯ ( r ) ) G ¯ Γ N , 2 ( m 0 , r ) e i k 0 r m 0 u ¯ ( m 0 ) n k 0 2 ( 1 + v ( r ) ) G Γ N , 3 ( m 0 , r ) + u ( m 0 ) e i k 0 r m 0 k 0 2 ( 1 + i k 0 cos ϕ ) ( 1 + v ¯ ( r ) ) G ¯ Γ N , 4 ( m 0 , r ) .
f ( x , y ) = { 0.001 if ( x 0.025 ) 2 + ( y 0.04 ) 2 0.008 0.001 i if ( x 0.055 ) 2 + ( y 0.04 ) 2 0.008 0 otherwise .
v δ ( r ) + 2 i k 0 θ v δ ( r ) k 0 2 v δ ( r ) f ( r ) = k 0 2 ( 1 + v ( r ) ) d ( r )
v δ ( r ) = 0 on L L , v δ ( r ) + v δ ( r ) n = 0 on L + .
v δ ( r ) = Ω [ G v ( r , r ) k 0 2 ( 1 + v ( r ) ) d ( r ) ] d n r .
Ω ψ ¯ [ v δ + 2 i k 0 θ v δ k 0 2 v δ f ] d n r = Ω ψ ¯ [ k 0 2 ( 1 + v ) d ] d n r .
Ω ψ ¯ [ v δ ] d n r = Ω v δ [ ψ ¯ ] d n r Ω v δ ψ ¯ n d n 1 r .
Ω ( 2 i k 0 θ v δ ) ψ ¯ d n r = Ω ( 2 i k 0 θ ψ ¯ ) v δ d n r + Ω 2 i k 0 v δ ψ ¯ θ n d n 1 r .
Ω ψ ¯ [ k 0 2 ( 1 + v ) d ] d n r = Ω v δ [ ψ ¯ 2 i k 0 θ ψ ¯ k 0 2 ψ ¯ f ] d n r Ω ψ ¯ n v δ d n 1 r + Ω v δ n ψ ¯ d n 1 r + Ω ( 2 i k 0 θ n ) v δ ψ ¯ d n 1 r .
ψ + 2 i k 0 θ ψ k 0 2 ψ f = 0 on Ω
q + ψ ( r ) n + ψ ( r ) ( 1 + 2 i k 0 θ n ) on L + ,
ψ ( r ) = 0 on L L ,
L + q + ¯ v δ d r n 1 = Ω ψ ¯ [ k 0 2 ( 1 + v ) d ] d n r .
Ω ψ ¯ [ k 0 2 ( 1 + v ) d ] d n r = v δ ( m 0 ) , m 0 L + .
v δ d ( r ) = G ¯ a d j ( v ) k 0 2 [ 1 + v ( r ) ] , r Ω .

Metrics