Abstract

We consider the problem of imaging the optical properties of a highly scattering medium probed by diffuse light. An analytic solution to this problem is derived from the singular value decomposition of the forward-scattering operator, which leads to explicit inversion formulas for the inverse scattering problem with diffusing waves. Computer simulations are used to illustrate these results in model systems.

© 2001 Optical Society of America

Full Article  |  PDF Article

Errata

John C. Schotland and Vadim A. Markel, "Inverse scattering with diffusing waves: erratum," J. Opt. Soc. Am. A 19, 1035-1035 (2002)
https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-19-5-1035

References

  • View by:
  • |
  • |
  • |

  1. G. J. Mueller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. van der Zee, eds., Medical Optical Tomography: Functional Imaging and Monitoring (SPIE Press, Bellingham, Wash., 1993).
  2. R. R. Alfano, J. G. Fujimoto, eds., Advances in Optical Imaging and Photon Migration, Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996).
  3. B. Chance, R. R. Alfano, eds., Optical Tomography and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, Proc. SPIE2389 (1995).
  4. B. Chance, R. R. Alfano, eds., Optical Tomography and Spectroscopy of Tissue and Model Media: Theory, Instrumentation, and Human Studies II, Proc. SPIE2949 (1997).
  5. B. Chance, R. R. Alfano, B. Tromberg, eds, Optical Tomography and Spectroscopy of Tissue III, Proc. SPIE3597, (1999).
  6. M. C. W. van Rossum, T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999).
    [CrossRef]
  7. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
    [CrossRef]
  8. L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
    [CrossRef] [PubMed]
  9. D. A. Benaron, D. K. Stevenson, “Optical time-of-flight and absorbance imaging of biologic media,” Science 259, 1463–1466 (1993).
    [CrossRef] [PubMed]
  10. A. Rebane, J. Feinberg, “Time-resolved holography,” Nature (London) 351, 378–380 (1991).
    [CrossRef]
  11. K. M. Yoo, F. Liu, R. R. Alfano, “Imaging objects hidden in scattering media using an absorption technique,” Opt. Lett. 16, 1068–1070 (1991).
    [CrossRef] [PubMed]
  12. E. Leith, H. Chen, Y. Chen, D. Dilworth, J. Lopez, R. Masri, J. Rudd, J. Valdmanis, “Electronic holography and speckle methods for imaging through tissue using femtosecond gated pulses,” Appl. Opt. 30, 4204–4210 (1991).
    [CrossRef] [PubMed]
  13. E. N. Leith, B. G. Hoover, D. S. Dilworth, P. P. Naulleau, “Ensemble-averaged Shack–Hartmann wavefront sensing for imaging through turbid media,” Appl. Opt. 37, 3643–3650 (1998).
    [CrossRef]
  14. J. R. Singer, F. A. Grunbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
    [CrossRef] [PubMed]
  15. S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “New results for the development of infrared absorption imaging,” in Biomedical Image Processing, A. C. Bovik, W. E. Higgins, eds., Proc. SPIE1245, 92–103 (1991).
    [CrossRef]
  16. R. L. Barbour, H. L. Graber, R. Aronson, J. Lubowsky, “Imaging of subsurface regions of random media by remote sensing,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzin, eds., Proc. SPIE1431, 192–203 (1991).
    [CrossRef]
  17. J. Schotland, J. Leigh, “Photon diffusion imaging,” Biophys. J. 61, 446 (1992).
  18. C. P. Gonatas, M. Ishii, J. S. Leigh, J. C. Schotland, “Optical diffusion imaging using a direct inversion method,” Phys. Rev. E 52, 4361–4365 (1995).
    [CrossRef]
  19. M. Ishii, J. Leigh, J. Schotland, “Photon diffusion imaging of model and biological systems,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human, Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 312–317 (1995).
  20. M. O’Leary, D. Boas, B. Chance, A. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusing photon tomography,” Opt. Lett. 20, 426–429 (1995).
    [CrossRef]
  21. J. C. Schotland, “Continuous wave diffusion imaging,” J. Opt. Soc. Am. A 14, 275–279 (1997).
    [CrossRef]
  22. X. D. Li, T. Durduran, A. Yodh, B. Chance, D. Pattanayak, “Diffraction tomography for biochemical imaging with diffuse photon density waves,” Opt. Lett. 22, 573 (1997).
    [CrossRef] [PubMed]
  23. C. Matson, “A diffraction tomographic model of the forward problem using photon density waves,” Opt. Express 1, 6–11 (1997).
    [CrossRef] [PubMed]
  24. F. Natterer, The Mathematics of Computerized Tomography (Wiley, New York, 1986).

1999

M. C. W. van Rossum, T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999).
[CrossRef]

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
[CrossRef]

1998

1997

1995

C. P. Gonatas, M. Ishii, J. S. Leigh, J. C. Schotland, “Optical diffusion imaging using a direct inversion method,” Phys. Rev. E 52, 4361–4365 (1995).
[CrossRef]

M. O’Leary, D. Boas, B. Chance, A. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusing photon tomography,” Opt. Lett. 20, 426–429 (1995).
[CrossRef]

1993

D. A. Benaron, D. K. Stevenson, “Optical time-of-flight and absorbance imaging of biologic media,” Science 259, 1463–1466 (1993).
[CrossRef] [PubMed]

1992

J. Schotland, J. Leigh, “Photon diffusion imaging,” Biophys. J. 61, 446 (1992).

1991

1990

J. R. Singer, F. A. Grunbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

Alfano, R. R.

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

K. M. Yoo, F. Liu, R. R. Alfano, “Imaging objects hidden in scattering media using an absorption technique,” Opt. Lett. 16, 1068–1070 (1991).
[CrossRef] [PubMed]

Aronson, R.

R. L. Barbour, H. L. Graber, R. Aronson, J. Lubowsky, “Imaging of subsurface regions of random media by remote sensing,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzin, eds., Proc. SPIE1431, 192–203 (1991).
[CrossRef]

Arridge, S. R.

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
[CrossRef]

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “New results for the development of infrared absorption imaging,” in Biomedical Image Processing, A. C. Bovik, W. E. Higgins, eds., Proc. SPIE1245, 92–103 (1991).
[CrossRef]

Barbour, R. L.

R. L. Barbour, H. L. Graber, R. Aronson, J. Lubowsky, “Imaging of subsurface regions of random media by remote sensing,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzin, eds., Proc. SPIE1431, 192–203 (1991).
[CrossRef]

Benaron, D. A.

D. A. Benaron, D. K. Stevenson, “Optical time-of-flight and absorbance imaging of biologic media,” Science 259, 1463–1466 (1993).
[CrossRef] [PubMed]

Boas, D.

Chance, B.

Chen, H.

Chen, Y.

Cope, M.

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “New results for the development of infrared absorption imaging,” in Biomedical Image Processing, A. C. Bovik, W. E. Higgins, eds., Proc. SPIE1245, 92–103 (1991).
[CrossRef]

Delpy, D. T.

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “New results for the development of infrared absorption imaging,” in Biomedical Image Processing, A. C. Bovik, W. E. Higgins, eds., Proc. SPIE1245, 92–103 (1991).
[CrossRef]

Dilworth, D.

Dilworth, D. S.

Durduran, T.

Feinberg, J.

A. Rebane, J. Feinberg, “Time-resolved holography,” Nature (London) 351, 378–380 (1991).
[CrossRef]

Gonatas, C. P.

C. P. Gonatas, M. Ishii, J. S. Leigh, J. C. Schotland, “Optical diffusion imaging using a direct inversion method,” Phys. Rev. E 52, 4361–4365 (1995).
[CrossRef]

Graber, H. L.

R. L. Barbour, H. L. Graber, R. Aronson, J. Lubowsky, “Imaging of subsurface regions of random media by remote sensing,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzin, eds., Proc. SPIE1431, 192–203 (1991).
[CrossRef]

Grunbaum, F. A.

J. R. Singer, F. A. Grunbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

Ho, P. P.

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

Hoover, B. G.

Ishii, M.

C. P. Gonatas, M. Ishii, J. S. Leigh, J. C. Schotland, “Optical diffusion imaging using a direct inversion method,” Phys. Rev. E 52, 4361–4365 (1995).
[CrossRef]

M. Ishii, J. Leigh, J. Schotland, “Photon diffusion imaging of model and biological systems,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human, Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 312–317 (1995).

Kohn, P.

J. R. Singer, F. A. Grunbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

Leigh, J.

J. Schotland, J. Leigh, “Photon diffusion imaging,” Biophys. J. 61, 446 (1992).

M. Ishii, J. Leigh, J. Schotland, “Photon diffusion imaging of model and biological systems,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human, Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 312–317 (1995).

Leigh, J. S.

C. P. Gonatas, M. Ishii, J. S. Leigh, J. C. Schotland, “Optical diffusion imaging using a direct inversion method,” Phys. Rev. E 52, 4361–4365 (1995).
[CrossRef]

Leith, E.

Leith, E. N.

Li, X. D.

Liu, C.

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

Liu, F.

Lopez, J.

Lubowsky, J.

R. L. Barbour, H. L. Graber, R. Aronson, J. Lubowsky, “Imaging of subsurface regions of random media by remote sensing,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzin, eds., Proc. SPIE1431, 192–203 (1991).
[CrossRef]

Masri, R.

Matson, C.

Natterer, F.

F. Natterer, The Mathematics of Computerized Tomography (Wiley, New York, 1986).

Naulleau, P. P.

Nieuwenhuizen, T. M.

M. C. W. van Rossum, T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999).
[CrossRef]

O’Leary, M.

Pattanayak, D.

Rebane, A.

A. Rebane, J. Feinberg, “Time-resolved holography,” Nature (London) 351, 378–380 (1991).
[CrossRef]

Rudd, J.

Schotland, J.

J. Schotland, J. Leigh, “Photon diffusion imaging,” Biophys. J. 61, 446 (1992).

M. Ishii, J. Leigh, J. Schotland, “Photon diffusion imaging of model and biological systems,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human, Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 312–317 (1995).

Schotland, J. C.

J. C. Schotland, “Continuous wave diffusion imaging,” J. Opt. Soc. Am. A 14, 275–279 (1997).
[CrossRef]

C. P. Gonatas, M. Ishii, J. S. Leigh, J. C. Schotland, “Optical diffusion imaging using a direct inversion method,” Phys. Rev. E 52, 4361–4365 (1995).
[CrossRef]

Singer, J. R.

J. R. Singer, F. A. Grunbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

Stevenson, D. K.

D. A. Benaron, D. K. Stevenson, “Optical time-of-flight and absorbance imaging of biologic media,” Science 259, 1463–1466 (1993).
[CrossRef] [PubMed]

Valdmanis, J.

van der Zee, P.

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “New results for the development of infrared absorption imaging,” in Biomedical Image Processing, A. C. Bovik, W. E. Higgins, eds., Proc. SPIE1245, 92–103 (1991).
[CrossRef]

van Rossum, M. C. W.

M. C. W. van Rossum, T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999).
[CrossRef]

Wang, L.

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

Yodh, A.

Yoo, K. M.

Zhang, G.

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

Zubelli, J. P.

J. R. Singer, F. A. Grunbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

Appl. Opt.

Biophys. J.

J. Schotland, J. Leigh, “Photon diffusion imaging,” Biophys. J. 61, 446 (1992).

Inverse Probl.

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
[CrossRef]

J. Opt. Soc. Am. A

Nature (London)

A. Rebane, J. Feinberg, “Time-resolved holography,” Nature (London) 351, 378–380 (1991).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. E

C. P. Gonatas, M. Ishii, J. S. Leigh, J. C. Schotland, “Optical diffusion imaging using a direct inversion method,” Phys. Rev. E 52, 4361–4365 (1995).
[CrossRef]

Rev. Mod. Phys.

M. C. W. van Rossum, T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999).
[CrossRef]

Science

J. R. Singer, F. A. Grunbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

D. A. Benaron, D. K. Stevenson, “Optical time-of-flight and absorbance imaging of biologic media,” Science 259, 1463–1466 (1993).
[CrossRef] [PubMed]

Other

G. J. Mueller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. van der Zee, eds., Medical Optical Tomography: Functional Imaging and Monitoring (SPIE Press, Bellingham, Wash., 1993).

R. R. Alfano, J. G. Fujimoto, eds., Advances in Optical Imaging and Photon Migration, Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996).

B. Chance, R. R. Alfano, eds., Optical Tomography and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, Proc. SPIE2389 (1995).

B. Chance, R. R. Alfano, eds., Optical Tomography and Spectroscopy of Tissue and Model Media: Theory, Instrumentation, and Human Studies II, Proc. SPIE2949 (1997).

B. Chance, R. R. Alfano, B. Tromberg, eds, Optical Tomography and Spectroscopy of Tissue III, Proc. SPIE3597, (1999).

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “New results for the development of infrared absorption imaging,” in Biomedical Image Processing, A. C. Bovik, W. E. Higgins, eds., Proc. SPIE1245, 92–103 (1991).
[CrossRef]

R. L. Barbour, H. L. Graber, R. Aronson, J. Lubowsky, “Imaging of subsurface regions of random media by remote sensing,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzin, eds., Proc. SPIE1431, 192–203 (1991).
[CrossRef]

M. Ishii, J. Leigh, J. Schotland, “Photon diffusion imaging of model and biological systems,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human, Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 312–317 (1995).

F. Natterer, The Mathematics of Computerized Tomography (Wiley, New York, 1986).

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

Experimental geometry. The sample occupies the half-space z>0 of an infinite medium with sources and detectors located on the plane z=0 with transverse coordinate ρ. A typical photon path is also shown.

Fig. 2
Fig. 2

Diagrammatic expansions of T and ln T. The solid lines indicate a factor of G0, the black circles represent a factor of δα weighted by the appropriate combinatorial factors, with the internal coordinates integrated, and the white circles indicate an unintegrated coordinate.

Fig. 3
Fig. 3

Contour plots of (a) ΓA and (b) ΓD in an infinite nonabsorbing medium for k0L=1, where L denotes the source–detector pair separation.

Fig. 4
Fig. 4

Tomographic images of four pointlike absorbers. The longitudinal ranges of the tomographic slices (all lengths are in centimeters) are -10x, y10. The coordinates of absorbers are x=-5, y=-5,z=2 (lower left corner); x=0, y=0, z=5 (center); x=5, y=-5, z=5 (lower right corner); and x=5, y=5, z=8 (upper right corner). The reconstruction is done for zero modulation frequency (ω=0) with wavelength λ=2π/k0=10. The images are calibrated from 0 (black) to 1 (white) in absolute scale. Here black corresponds to δα=0 and white to the maximum absolute value of δα obtained for a pointlike absorber located at a particular depth.

Fig. 5
Fig. 5

Reconstruction in a one-dimensional geometry with infinitely thin plane absorbers located at different depths z0.

Fig. 6
Fig. 6

Simultaneous reconstruction of δα and δD for one point absorber and one point scatterer, both located at the depth z=2 (all lengths are in centimeters). The absorber is located in the lower left corner (x=-5, y=-5, z=2); the scatterer is located in the upper right corner (x=5, y=5, z=2). Other parameters are the same as those in Fig. 4. The images for δα are not calibrated, since reconstructed δα has no maxima in the z direction; calibration of δD images is done as in Fig. 4.

Fig. 7
Fig. 7

Reconstruction in a one-dimensional geometry for different infinitely thin absorbers and scatterers: (a) one absorber at z=2 cm; (b) one scatterer at z=2 cm; (c) one absorber and one scatterer, both at z=2 cm; (d) one absorber at z=2 cm and one scatterer at z=4 cm.

Equations (80)

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

tu(r, t)=·[D(r)u(r, t)]-α(r)u(r, t)+S(r, t),
S(r, t)=[1+A exp(-iωt)]S(r),
u(r, t)=u0(r)+A exp(-iωt)u(r),
-·[D(r)u(r)]+[α(r)-iω]u(r)=S(r).
u(r)=d3r G(r, r)S(r),
[-r·D(r)r+α(r)-iω]G(r, r)=δ(r-r).
G0(r1, r2)=exp(-k0|r1-r2|)4πD0|r1-r2|,
k02=α0-iωD0.
 G0(r, r)=12D0  d2q(2π)2Q(q)-1×exp[iq·(ρ-ρ)-Q(q)|z-z|],
G0(r, r)=1D0  d3k(2π)3 exp[ik·(r-r)]k2+k02,
- exp(ikzz)kz2+q2+k02 dkz=πQ(q) exp[-Q(q)|z|],
Q(q)(q2+k02)1/2.
G(r1, r2)=G0(r1, r2)-d3r[G0(r1, r)G(r, r2)δα(r)+rG0(r1, r)·rG(r, r2)δD(r)],
T(r1, r2)=1-1G0(r1, r2)×d3r[G0(r1, r)G(r, r2)δα(r)+rG0(r1, r)·rG(r, r2)δD(r)].
-ln T(r1, r2)=d3r[ΓA(r;r1, r2)δα(r)+ΓD(r;r1, r2)δD(r)],
ΓA(r;r1, r2)=-δ ln T(r1, r2)δα(r)δα(r)=0,δD(r)=0=1G0(r1, r2)G0(r1, r)G0(r, r2)
ΓD(r;r1, r2)=-δ ln T(r1, r2)δD(r)δα(r)=0,δD(r)=0=1G0(r1, r2)rG0(r1, r)·rG0(r,r2).
ϕ(ρ1, ρ2)=d3r[G0(ρ1, 0;r)G0(ρ2, 0;r)δα(r)+rG0(ρ1, 0;r)·rG0(ρ2, 0;r)δD(r)].
ϕq(ρ1)=d2ρ2 exp(iq·ρ2)ϕ(ρ1, ρ2).
ϕq(ρ1)=d3r[KqA(ρ1, r)δα(r)+KqD(ρ1, r)δD(r)],
KqA(ρ1, r)=12D0Q(q) exp[iq·ρ-Q(q)|z|]×G0(ρ1, 0;r),
KqD(ρ1, r)=12D0Q(q) exp[iq·ρ-Q(q)|z|]×[iq-Q(q)zˆ]·rG0(ρ1, 0;r)
ϕq(ρ1)=d3r KqA(ρ1, r)δα(r).
KqA(ρ1, r)=d2q σqqAgqq(ρ1)fqqA*(r),
gqq(ρ)=12π exp[i(q+q)·ρ],
fqqA(r)=Nqq2π exp(i(q·ρ+θqqA)-{[Q(q)]*+[Q(q+q)]*}|z|),
σqqA=14D02Nqq|Q(q)Q(q+q)|.
exp(iθqqA)=Q(q)Q(q+q)|Q(q)Q(q+q)|.
KqA*KqAfqqA=(σqqA)2fqqA,
KqAfqqA=σqqAgqq.
δα(r)=d2ρ1 KqA+(r, ρ1)ϕq(ρ1),
KqA+(r, ρ1)=d2q 1σqqAgqq*(ρ1)fqqA(r).
ϕq(ρ1)=d3r KqD(ρ1, r)δD(r).
KqD(ρ1, r)=d2q σqqDgqq(ρ1)fqqD*(r)
σqqD=σqqA|q·(q+q)+Q(q)Q(q+q)|
exp(iθqqD)=q·(q+q)+Q(q)Q(q+q)|q·(q+q)+Q(q)Q(q+q)|×exp(iθqqA).
δD(r)=d2ρ1 KqD+(r, ρ1)ϕq(ρ1),
KqD+(r, ρ1)=d2q 1σqqDgqq*(ρ1)fqqD(r).
δα=KqA*(KqAKqA*+KqDKqD*)-1ϕq,
δD=KqD*(KqAKqA*+KqDKqD*)-1ϕq.
δα(r)=d2ρ1d2q σqqA[(σqqA)2+(σqqD)2]-1×gqq*(ρ1)fqqA(r)ϕq(ρ1),
δD(r)=d2ρ1d2q σqqD[(σqqA)2+(σqqD)2]-1×gqq*(ρ1)fqqD(r)ϕq(ρ1),
KqiAδα+KqiDδD=ϕqi
Kf=ϕ,
K=Kq1AKqiDKqpAKqpD,
K(ρ1,r)=d2ql=1pσqlϕql(ρ1)ψql*(r).
KK*ϕql=σql2ϕql,
K*ϕql=σqlψql.
j=1p(KqiAKqjA*+KqiDKqjD*)ϕqlj=σql2ϕqli.
KqiAKqjA*gqqj=χijA(q)σqqiAσqqjAgqqi,
KqiDKqjD*qqqj=χijD(q)σqqiDσqqjDgqqi.
fqqiA,fqqjA=χijA(q)δ(q-q),
fqqiD,fqqjD=χijD(q)δ(q-q).
jMij(q)clj(q)=σql2cli(q),
Mij(q)=χijA(q)σqqiAσqqjA+χijD(q)σqqiDσqqjd.
ψqlA(r)=1σql jσqqjAclj(q)fqqjA(r),
ψqlD(r)=1σql jσqqjDclj(q)fqqjD(r).
f(r)=d2ρ1 K+(r, ρ1)ϕ(ρ1),
K+(r, ρ1)=d2ql 1σqlψql(r)ϕql*(ρ1).
δα(r)=d2ρ1d2ql 1σqlψqlA(r)ϕql*(ρ1)ϕ(ρ1),
δD(r)=d2ρ1d2ql 1σqlψqlD(r)ϕql*(ρ1)ϕ(ρ1),
Kqif=ϕqi,
fn+1=iPifn,
Pif=f+Kqi*(KqiKqi*)-1(ϕqi-Kqif).
fni+1=fni+Kqi*(KqiKqi*)-1(ϕqi-Kqifni),
δαni+1=δαni+KqiA*(KqiAKqiA*+KqiDKqiD*)-1×(ϕqi-KqiAδαni-KqiDδDni),
δDni+1=δDni+KqiD*(KqiAKqiA*+KqiDKqiD*)-1×(ϕqi-KqiAδαni-KqiDδDni).
δαni+1(r)=δαni(r)+d2ρ1d2q σqqiA[(σqqiA)2+(σqqiD)2]-1gqqi*(ρ1)fqqiA(r)[ϕqi(ρ1)-KqiAδαni(ρ1)-KqiDδDni(ρ1)],
δDni+1(r)=δDni(r)+d2ρ1d2q σqqiD[(σqqiA)2+(σqqiD)2]-1gqqi*(ρ1)fqqiD(r)[ϕqi(ρ1)-KqiAδαni(ρ1)-KqiDδDni(ρ1)],
δα(r)=D0π2d2q exp(iq·ρ)×k,lηA(z;q, qk)Akl-1(q)ζ(q, ql),
δD(r)=D0π2d2q exp(iq·ρ)×k,lηD(z;q, qk)Akl-1(q)ζ(q, ql),
ηA(z;q, qk)=exp{-[Q*(qk)+Q*(q+qk)]z},
ηD(z;q, qk)=t*(qk, q)exp{-[Q*(qk)+Q*(q+qk)]z},
ζ(q, qk)=Q(qk)Q(qk+q)ϕˆ(-(qk+q), qk),
Akl(q)=1-exp{[Q(qk)+Q(qk+q)+Q*(ql)+Q*(ql+q)]L}Q(qk)+Q(qk+q)+Q*(ql)+Q*(ql+q)[1+t(qk, q)t*(ql, q)],
t(qk, q)=qk·(qk+q)+Q(qk)Q(qk+q),
δα(r)=k=1NAαkδ(r-rkA),
δD(r)=k=1NDDkδ(r-rkD).
ϕˆ(-(qk+q), qk)=1(2D0)2Q(qk)Q(qk+q)l=1NAαl exp{-iq·ρlA-[Q(qk)+Q(qk+q)]zlA}+t(qk, q)l=1NDDl×exp{-iq·ρlD-[Q(qk)+Q(qk+q)]zlD}.
ϕˆ(-(qk+q), qk)=α0πD02 δ(q)exp[-2Q(qk)]Q2(qk).

Metrics