Abstract

We present a formalism for solving the scalar Bethe–Salpeter equation (BSE) in the nondiffusive regime under the ladder approximation and for an infinite randomly scattering medium having scatterers of size on the order of or larger than the wavelength. We compare the information content in a wave transport model (the BSE) with that in energy-based transport, the Boltzmann transport equation (BTE), in the spatial frequency domain. Our results suggest that when absorption dominates scatter, the intensity Green’s function from a BTE model is similar to the field correlation Green’s function from a BSE solution. When scatter dominates loss, there are significant differences between the BTE and BSE representations, and the BTE solutions appear to be smoothed versions of those from the BSE. Therefore, field correlation measurements, perhaps extracted from intensity correlations over frequency and space, offer significantly more information than a mean-intensity measurement in the weakly scattering and nondiffusive regime. Our work provides a mathematical framework for electric field correlation-based imaging methods based on the BSE that hold promise in, for example, near-surface tissue imaging.

© 2013 Optical Society of America

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References

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    [CrossRef]
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2007 (1)

H. Subramanian, P. Pradhan, Y. Kim, and V. Backman, “Penetration depth of low-coherence enhanced backscattered light in subdiffusion regime,” Phys. Rev. E 75, 41914 (2007).
[CrossRef]

2006 (1)

2005 (2)

Y. Kim, Y. Liu, V. Turzhitsky, R. Wali, H. Roy, and V. Backman, “Depth-resolved low-coherence enhanced backscattering,” Opt. Lett. 30, 741–743 (2005).
[CrossRef]

A. Lubatsch, J. Kroha, and K. Busch, “Theory of light diffusion in disordered media with linear absorption or gain,” Phys. Rev. B 71, 184201 (2005).
[CrossRef]

2004 (4)

2002 (3)

A. B. Milstein, S. Oh, J. S. Reynolds, K. J. Webb, C. A. Bouman, and R. P. Millane, “Three-dimensional Bayesian optical diffusion tomography with experimental data,” Opt. Lett. 27, 95–97 (2002).
[CrossRef]

X. Zhang and Z.-Q. Zhang, “Wave transport through thin slabs of random media with internal reflection: ballistic to diffusive transition,” Phys. Rev. E 66, 016612 (2002).
[CrossRef]

A. D. Klose and A. H. Hielscher, “Optical tomography using the time-independent equation of radiative transfer-part 2: inverse model,” J. Quant. Spectrosc. Radiat. Transfer 72, 715–732 (2002).
[CrossRef]

1999 (3)

J. M. Schmitt, “Optical coherence tomography (OCT): a review,” IEEE J. Sel. Top. Quantum Electron. 5, 1205–1215 (1999).
[CrossRef]

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, “Wave transport in random media: the ballistic to diffusive transition,” Phys. Rev. E 60, 4843–4850 (1999).
[CrossRef]

A. Hielscher and A. Klose, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phys. 26, 1698 (1999).
[CrossRef]

1997 (1)

S. R. Arridge and J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef]

1996 (5)

A. Lagendijk and B. Tiggelen, “Resonant multiple scattering of light,” Phys. Rep. 270, 143–215 (1996).
[CrossRef]

V. S. Podolsky and A. A. Lisyansky, “Transfer matrix of a spherical scatterer,” Phys. Rev. B 54, 12125–12128 (1996).
[CrossRef]

D. Livdan and A. Lisyanki, “Transport properties of waves in absorbing random media with microstructure,” Phys. Rev. B 53, 14843 (1996).
[CrossRef]

S. John, G. Pang, and Y. Yang, “Optical coherence propagation and imaging in a multiple scattering medium,” J. Biomed. Opt. 1, 180–191 (1996).
[CrossRef]

H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, and M. S. Patterson, “Optical image reconstruction using frequency domain data: simulations and experiments,” J. Opt. Soc. Am. A 13, 253–266 (1996).
[CrossRef]

1991 (1)

D. Huang, E. Swanson, C. Lin, J. Schuman, W. Stinson, W. Chang, M. Hee, T. Flotte, D. Gregory, C. Puliafito, and J. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef]

1988 (1)

M. B. van der Mark, M. P. van Albada, and A. Lagendijk, “Light scattering in strongly scattering media: multiple scattering and weak localization,” Phys. Rev. B 37, 3575–3592 (1988).
[CrossRef]

1962 (1)

I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. Theory 8, 194–195 (1962).
[CrossRef]

1951 (1)

E. Salpeter and H. Bethe, “A relativistic equation for bound-state problems,” Phys. Rev. 84, 1232 (1951).
[CrossRef]

Akkermans, E.

E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University, 2007).

Arridge, S. R.

S. R. Arridge and J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef]

Backman, V.

Bethe, H.

E. Salpeter and H. Bethe, “A relativistic equation for bound-state problems,” Phys. Rev. 84, 1232 (1951).
[CrossRef]

Boas, D. A.

Bouman, C. A.

Busch, K.

A. Lubatsch, J. Kroha, and K. Busch, “Theory of light diffusion in disordered media with linear absorption or gain,” Phys. Rev. B 71, 184201 (2005).
[CrossRef]

Chang, W.

D. Huang, E. Swanson, C. Lin, J. Schuman, W. Stinson, W. Chang, M. Hee, T. Flotte, D. Gregory, C. Puliafito, and J. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef]

Duderstadt, J. J.

J. J. Duderstadt and L. J. Hamilton, Nuclear Reactor Analysis (Wiley, 1976).

Florescu, L.

L. Florescu and S. John, “Theory of photon statistics and optical coherence in a multiple-scattering random-laser medium,” Phys. Rev. E 69, 046603 (2004).
[CrossRef]

Flotte, T.

D. Huang, E. Swanson, C. Lin, J. Schuman, W. Stinson, W. Chang, M. Hee, T. Flotte, D. Gregory, C. Puliafito, and J. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef]

Frisch, U.

U. Frisch, “Wave propagation in random media,” in Probability Methods in Applied Mathematics, A. T. Bharucha-Reid, ed. (Academic, 1968), Vol. 1, pp. 76–198.

Fujimoto, J.

D. Huang, E. Swanson, C. Lin, J. Schuman, W. Stinson, W. Chang, M. Hee, T. Flotte, D. Gregory, C. Puliafito, and J. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef]

Gerke, T. D.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 1985).

Gregory, D.

D. Huang, E. Swanson, C. Lin, J. Schuman, W. Stinson, W. Chang, M. Hee, T. Flotte, D. Gregory, C. Puliafito, and J. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef]

Hamilton, L. J.

J. J. Duderstadt and L. J. Hamilton, Nuclear Reactor Analysis (Wiley, 1976).

Hebden, J. C.

S. R. Arridge and J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef]

Hee, M.

D. Huang, E. Swanson, C. Lin, J. Schuman, W. Stinson, W. Chang, M. Hee, T. Flotte, D. Gregory, C. Puliafito, and J. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef]

Hielscher, A.

A. Hielscher and A. Klose, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phys. 26, 1698 (1999).
[CrossRef]

Hielscher, A. H.

A. D. Klose and A. H. Hielscher, “Optical tomography using the time-independent equation of radiative transfer-part 2: inverse model,” J. Quant. Spectrosc. Radiat. Transfer 72, 715–732 (2002).
[CrossRef]

Huang, D.

D. Huang, E. Swanson, C. Lin, J. Schuman, W. Stinson, W. Chang, M. Hee, T. Flotte, D. Gregory, C. Puliafito, and J. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1977), Vol. 2.

Jiang, H.

John, S.

L. Florescu and S. John, “Theory of photon statistics and optical coherence in a multiple-scattering random-laser medium,” Phys. Rev. E 69, 046603 (2004).
[CrossRef]

S. John, G. Pang, and Y. Yang, “Optical coherence propagation and imaging in a multiple scattering medium,” J. Biomed. Opt. 1, 180–191 (1996).
[CrossRef]

Jones, I. P.

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, “Wave transport in random media: the ballistic to diffusive transition,” Phys. Rev. E 60, 4843–4850 (1999).
[CrossRef]

Kim, Y.

Klose, A.

A. Hielscher and A. Klose, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phys. 26, 1698 (1999).
[CrossRef]

Klose, A. D.

A. D. Klose and A. H. Hielscher, “Optical tomography using the time-independent equation of radiative transfer-part 2: inverse model,” J. Quant. Spectrosc. Radiat. Transfer 72, 715–732 (2002).
[CrossRef]

Kroha, J.

A. Lubatsch, J. Kroha, and K. Busch, “Theory of light diffusion in disordered media with linear absorption or gain,” Phys. Rev. B 71, 184201 (2005).
[CrossRef]

Lagendijk, A.

A. Lagendijk and B. Tiggelen, “Resonant multiple scattering of light,” Phys. Rep. 270, 143–215 (1996).
[CrossRef]

M. B. van der Mark, M. P. van Albada, and A. Lagendijk, “Light scattering in strongly scattering media: multiple scattering and weak localization,” Phys. Rev. B 37, 3575–3592 (1988).
[CrossRef]

Lin, C.

D. Huang, E. Swanson, C. Lin, J. Schuman, W. Stinson, W. Chang, M. Hee, T. Flotte, D. Gregory, C. Puliafito, and J. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef]

Lisyanki, A.

D. Livdan and A. Lisyanki, “Transport properties of waves in absorbing random media with microstructure,” Phys. Rev. B 53, 14843 (1996).
[CrossRef]

Lisyansky, A. A.

V. S. Podolsky and A. A. Lisyansky, “Transfer matrix of a spherical scatterer,” Phys. Rev. B 54, 12125–12128 (1996).
[CrossRef]

Liu, Y.

Livdan, D.

D. Livdan and A. Lisyanki, “Transport properties of waves in absorbing random media with microstructure,” Phys. Rev. B 53, 14843 (1996).
[CrossRef]

Lubatsch, A.

A. Lubatsch, J. Kroha, and K. Busch, “Theory of light diffusion in disordered media with linear absorption or gain,” Phys. Rev. B 71, 184201 (2005).
[CrossRef]

Millane, R. P.

Milstein, A. B.

Montambaux, G.

E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University, 2007).

Oh, S.

Osterberg, U. L.

Page, J. H.

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, “Wave transport in random media: the ballistic to diffusive transition,” Phys. Rev. E 60, 4843–4850 (1999).
[CrossRef]

Pang, G.

S. John, G. Pang, and Y. Yang, “Optical coherence propagation and imaging in a multiple scattering medium,” J. Biomed. Opt. 1, 180–191 (1996).
[CrossRef]

Patterson, M. S.

Paulsen, K. D.

Podolsky, V. S.

V. S. Podolsky and A. A. Lisyansky, “Transfer matrix of a spherical scatterer,” Phys. Rev. B 54, 12125–12128 (1996).
[CrossRef]

Pogue, B. W.

Pradhan, P.

H. Subramanian, P. Pradhan, Y. Kim, and V. Backman, “Penetration depth of low-coherence enhanced backscattered light in subdiffusion regime,” Phys. Rev. E 75, 41914 (2007).
[CrossRef]

Puliafito, C.

D. Huang, E. Swanson, C. Lin, J. Schuman, W. Stinson, W. Chang, M. Hee, T. Flotte, D. Gregory, C. Puliafito, and J. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef]

Reed, I. S.

I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. Theory 8, 194–195 (1962).
[CrossRef]

Reynolds, J. S.

Roy, H.

Salpeter, E.

E. Salpeter and H. Bethe, “A relativistic equation for bound-state problems,” Phys. Rev. 84, 1232 (1951).
[CrossRef]

Schmitt, J. M.

J. M. Schmitt, “Optical coherence tomography (OCT): a review,” IEEE J. Sel. Top. Quantum Electron. 5, 1205–1215 (1999).
[CrossRef]

Schriemer, H. P.

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, “Wave transport in random media: the ballistic to diffusive transition,” Phys. Rev. E 60, 4843–4850 (1999).
[CrossRef]

Schuman, J.

D. Huang, E. Swanson, C. Lin, J. Schuman, W. Stinson, W. Chang, M. Hee, T. Flotte, D. Gregory, C. Puliafito, and J. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef]

Sheng, P.

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, “Wave transport in random media: the ballistic to diffusive transition,” Phys. Rev. E 60, 4843–4850 (1999).
[CrossRef]

P. Sheng, Introduction to Wave Scattering, Localization and Mesoscopic Phenomena (Springer, 2006).

Stinson, W.

D. Huang, E. Swanson, C. Lin, J. Schuman, W. Stinson, W. Chang, M. Hee, T. Flotte, D. Gregory, C. Puliafito, and J. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef]

Stott, J. J.

Subramanian, H.

H. Subramanian, P. Pradhan, Y. Kim, and V. Backman, “Penetration depth of low-coherence enhanced backscattered light in subdiffusion regime,” Phys. Rev. E 75, 41914 (2007).
[CrossRef]

Swanson, E.

D. Huang, E. Swanson, C. Lin, J. Schuman, W. Stinson, W. Chang, M. Hee, T. Flotte, D. Gregory, C. Puliafito, and J. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef]

Tiggelen, B.

A. Lagendijk and B. Tiggelen, “Resonant multiple scattering of light,” Phys. Rep. 270, 143–215 (1996).
[CrossRef]

Turzhitsky, V.

van Albada, M. P.

M. B. van der Mark, M. P. van Albada, and A. Lagendijk, “Light scattering in strongly scattering media: multiple scattering and weak localization,” Phys. Rev. B 37, 3575–3592 (1988).
[CrossRef]

van der Mark, M. B.

M. B. van der Mark, M. P. van Albada, and A. Lagendijk, “Light scattering in strongly scattering media: multiple scattering and weak localization,” Phys. Rev. B 37, 3575–3592 (1988).
[CrossRef]

Wali, R.

Wang, Z.

Webb, K. J.

Webster, M. A.

Weiner, A. M.

Weitz, D. A.

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, “Wave transport in random media: the ballistic to diffusive transition,” Phys. Rev. E 60, 4843–4850 (1999).
[CrossRef]

Yang, Y.

S. John, G. Pang, and Y. Yang, “Optical coherence propagation and imaging in a multiple scattering medium,” J. Biomed. Opt. 1, 180–191 (1996).
[CrossRef]

Zhang, X.

X. Zhang and Z.-Q. Zhang, “Wave transport through thin slabs of random media with internal reflection: ballistic to diffusive transition,” Phys. Rev. E 66, 016612 (2002).
[CrossRef]

Zhang, Z. Q.

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, “Wave transport in random media: the ballistic to diffusive transition,” Phys. Rev. E 60, 4843–4850 (1999).
[CrossRef]

Zhang, Z.-Q.

X. Zhang and Z.-Q. Zhang, “Wave transport through thin slabs of random media with internal reflection: ballistic to diffusive transition,” Phys. Rev. E 66, 016612 (2002).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

J. M. Schmitt, “Optical coherence tomography (OCT): a review,” IEEE J. Sel. Top. Quantum Electron. 5, 1205–1215 (1999).
[CrossRef]

IRE Trans. Inf. Theory (1)

I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. Theory 8, 194–195 (1962).
[CrossRef]

J. Biomed. Opt. (1)

S. John, G. Pang, and Y. Yang, “Optical coherence propagation and imaging in a multiple scattering medium,” J. Biomed. Opt. 1, 180–191 (1996).
[CrossRef]

J. Opt. Soc. Am. A (3)

J. Quant. Spectrosc. Radiat. Transfer (1)

A. D. Klose and A. H. Hielscher, “Optical tomography using the time-independent equation of radiative transfer-part 2: inverse model,” J. Quant. Spectrosc. Radiat. Transfer 72, 715–732 (2002).
[CrossRef]

Med. Phys. (1)

A. Hielscher and A. Klose, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phys. 26, 1698 (1999).
[CrossRef]

Opt. Lett. (4)

Phys. Med. Biol. (1)

S. R. Arridge and J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef]

Phys. Rep. (1)

A. Lagendijk and B. Tiggelen, “Resonant multiple scattering of light,” Phys. Rep. 270, 143–215 (1996).
[CrossRef]

Phys. Rev. (1)

E. Salpeter and H. Bethe, “A relativistic equation for bound-state problems,” Phys. Rev. 84, 1232 (1951).
[CrossRef]

Phys. Rev. B (4)

M. B. van der Mark, M. P. van Albada, and A. Lagendijk, “Light scattering in strongly scattering media: multiple scattering and weak localization,” Phys. Rev. B 37, 3575–3592 (1988).
[CrossRef]

V. S. Podolsky and A. A. Lisyansky, “Transfer matrix of a spherical scatterer,” Phys. Rev. B 54, 12125–12128 (1996).
[CrossRef]

D. Livdan and A. Lisyanki, “Transport properties of waves in absorbing random media with microstructure,” Phys. Rev. B 53, 14843 (1996).
[CrossRef]

A. Lubatsch, J. Kroha, and K. Busch, “Theory of light diffusion in disordered media with linear absorption or gain,” Phys. Rev. B 71, 184201 (2005).
[CrossRef]

Phys. Rev. E (4)

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, “Wave transport in random media: the ballistic to diffusive transition,” Phys. Rev. E 60, 4843–4850 (1999).
[CrossRef]

X. Zhang and Z.-Q. Zhang, “Wave transport through thin slabs of random media with internal reflection: ballistic to diffusive transition,” Phys. Rev. E 66, 016612 (2002).
[CrossRef]

H. Subramanian, P. Pradhan, Y. Kim, and V. Backman, “Penetration depth of low-coherence enhanced backscattered light in subdiffusion regime,” Phys. Rev. E 75, 41914 (2007).
[CrossRef]

L. Florescu and S. John, “Theory of photon statistics and optical coherence in a multiple-scattering random-laser medium,” Phys. Rev. E 69, 046603 (2004).
[CrossRef]

Science (1)

D. Huang, E. Swanson, C. Lin, J. Schuman, W. Stinson, W. Chang, M. Hee, T. Flotte, D. Gregory, C. Puliafito, and J. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef]

Other (6)

E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University, 2007).

P. Sheng, Introduction to Wave Scattering, Localization and Mesoscopic Phenomena (Springer, 2006).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1977), Vol. 2.

J. J. Duderstadt and L. J. Hamilton, Nuclear Reactor Analysis (Wiley, 1976).

U. Frisch, “Wave propagation in random media,” in Probability Methods in Applied Mathematics, A. T. Bharucha-Reid, ed. (Academic, 1968), Vol. 1, pp. 76–198.

J. W. Goodman, Statistical Optics (Wiley, 1985).

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Figures (3)

Fig. 1.
Fig. 1.

For the LS-HA-1 regime (see Table 1), we compare an analytical solution of the scalar BSE under a point scatterer representation [7] with our numerical result for the field correlation Green’s function for the scalar BSE, |G200(q;ν,ν+Δν)|, and the intensity Green’s function for the BTE, |G200(q;Δν)|, at [q^θ,q^ϕ]=[π/4,π/4], and Δν=5MHz. Notice that the numerical solution of the BSE and the BTE agree with the analytical result at low spatial frequencies, but that the BTE result differs for large |q|. The units for |q| are m1.

Fig. 2.
Fig. 2.

Comparison of the BSE and the BTE in the LS-HA-2 regime (see Table 1). In (a)–(f), we plot the field correlation Green’s function, |G2lm(q,qd=k0;ν,ν+Δν)|, for the BSE and the intensity Green’s function for the BTE, |G2lm(q;Δν)|, at [q^θ,q^ϕ]=[π/4,π/4], l=[0,1,2] and m=[0,1,2]. Note the similarity between the BSE and BTE curves. The results show that when absorption dominates scatter, the impact of the wave nature of field is minimal and therefore measuring the field correlation may not offer more information than mean-intensity measurements.

Fig. 3.
Fig. 3.

Comparison of the BSE and the BTE in the LS-LA regime (see Table 1). In (a)–(f) we plot the field correlation Green’s function, |G2lm(q,qd=k0;ν,ν+Δν)|, for the BSE and the intensity Green’s function for the BTE, |G2lm(q;Δν)| at [q^θ,q^ϕ]=[π/4,π/4], at l=[0,1,2] and m=[0,1,2]. Note the significant differences between the BSE and BTE curves. This result suggests that the BTE is a smoothed version of the BSE and that in a LS-LA random medium there is significant information available in a field correlation measurement as compared to a simple mean-intensity measurement.

Tables (1)

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Table 1. Scattering Regimesa

Equations (21)

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G(rd,rs;ν)=G0(rd,rs;ν)+k02VG0(rd,r;ν)(ε(r)εb)G(r,rs;ν)dr,
G(rd,rs;ν)=G0(rd,rs;ν)+k02VG0(rd,r;ν)(ε(r)εb)G(r,rs;ν)dr,
G(rd,rs;ν)=G0(rd,rs;ν)+VVG0(rd,r;ν)M(r,r;ν)G(r,rs;ν)drr,
M1(r,r)=j=1NVS(rRj,rRj;ν)p(Rj)dRj,
G(rd,rs;ν)=G0(rd,rs;ν)+j=1NVVV[G0(rd,r;ν)S(rRj,rRj;ν)×p(Rj)G(r,rs;ν)drdrdRj].
G(rd,rs;ν)=Aexp(ik|rdrs|)4π|rdrs|,
G(K;ν)=[k02K.K+ik0ls]1,
E(rd1;ν)E*(rd2;ν+Δν)=VVG2(rd1,rd2;rs1,rs2;ν,ν+Δν)Ei(rs1;ν)×Ei*(rs2;ν+Δν)drs1drs2,
G2(rd1,rd2;rs1,rs2;ν,ν+Δν)=G(rd1,rs1;ν)G*(rd2,rs2;ν+Δν)+VVVVG(rd1,r;ν)G*(rd2,r¯;ν+Δν)×U(r,r¯;r,r¯;ν,ν+Δν)×G2(r,r¯;rs1,rs2;ν,ν+Δν)drdr¯drdr¯,
U1(r,r¯;r,r¯;ν,ν+Δν)=j=1NVS(rRj,rRj;ν)×S*(r¯Rj,r¯Rj;ν+Δν)p(Rj)dRj.
G2(q,qd,qs;ν,ν+Δν)=G(qd+;ν).G*(qd;ν+Δν).[δqd,qs+nq1S(qd+;q1+;ν)S*(qd;q1;ν+Δν)×G2(q,q1,qs;ν,ν+Δν)dq1],
G(qd+;ν)G*(qd;ν+Δν)=ΔG(q,qd;ν,ν+Δν)ΔG˜1(q,qd;ν,ν+Δν).
ΔG˜1(q,qd;ν,ν+Δν).G2(q,qd;ν,ν+Δν)=ΔG(q,qd;ν,ν+Δν)×[1+nq1S(qd+;q1+;ν)S*(qd;q1;ν+Δν)×G2(q,q1;ν,ν+Δν)dq1].
ΔG˜1(·)={[2π(ν+Δν)nbc]2|qd|2i2π(ν+Δν)nbcls}[(2πνnbc)2|qd+|2+i2πνnbcls].
2ik0[i2πnbc(Δν+Δν22ν)+iqd.qk0+1+(Δν2ν)ls]G2(q,qd;ν,ν+Δν)=ΔG(q,qd;ν,ν+Δν).[1+nq1S(qd+;q1+;ν)×S*(qd;q1;ν+Δν)G2(q,q1;ν,ν+Δν)dq1].
G2(q,qj;ν,ν+Δν)G2(q,qj;ν,ν+Δν)δ(|qj|k0)
[i2πnbΔνc+iq^d.q+1ls]G2(q,q^d;ν,ν+Δν)=[1+μsq^1P(q^d,q^1)×G2(q,q^1;ν,ν+Δν)dq^1],
G2(q,qj;ν,ν+Δν)=l,mG2lm(q,qj;ν,ν+Δν)Ylm(q^j).
l1m1Al1m1l2m2(q,qd;ν,ν+Δν)G2l1m1(q,qd;ν,ν+Δν)=Bl2m2(q,qd;ν,ν+Δν)+l3m3q1q12Cl3m3l2m2(q,q1;ν,ν+Δν)G2l3m3(q,q1;ν,ν+Δν)dq1,
Al1m1l2m2(q,qd;ν,ν+Δν)=2ik0q^d[i2πnbc(Δν+Δν22ν)+iqd.qk0+1+(Δν2ν)ls]×Yl1m1(q^d)Y*l2m2(q^d)dq^dBl2m2(q,qd;ν,ν+Δν)=q^dΔG(q,qd;ν,ν+Δν)Y*l2m2(q^d)dq^dCl3m3l2m2(q,q1;ν,ν+Δν)=q^dq^1ΔG(q,qd;ν,ν+Δν)S(qd+;q1+;ν)S*(qd;q1;ν+Δν)×Y*l2m2(q^d)Yl3m3(q^1)dq^1dq^d.
G2lm(q,ν,ν+Δν)=k0Δk0+Δqd2G2lm(q,qd;ν,ν+Δν)dqd.

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