Abstract

Optical microscopy offers a unique insight of biological structures with a sub-micrometer resolution and a minimum invasiveness. However, the inhomogeneities of the specimen itself can induce multiple scattering of light and optical aberrations which limit the observation to depths close to the surface. To predict quantitatively the penetration depth in microscopy, we theoretically derive the single-to-multiple scattering ratio in reflection. From this key quantity, the multiple scattering limit is deduced for various microscopic imaging techniques such as confocal microscopy, optical coherence tomography and related methods.

© 2017 Optical Society of America

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References

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2017 (1)

2016 (1)

A. Badon, D. Li, G. Lerosey, A. C. Boccara, M. Fink, and A. Aubry, “Smart optical coherence tomography for ultra-deep imaging through highly scattering media,” Sci. Adv. 2, e1600370 (2016).
[Crossref] [PubMed]

2015 (3)

J. H. Park, W. Sun, and M. Cui, “High-resolution in vivo imaging of mouse brain through the intact skull,” Proc. Natl. Acad. Sci. U. S. A. 112, 9236–9241 (2015).
[Crossref] [PubMed]

S. Kang, S. Jeong, W. Choi, H. Ko, T. D. Yang, J. H. Joo, J.-S. Lee, Y.-S. Lim, Q. H. Park, and W. Choi, “Imaging deep within a scattering medium using collective accumulation of single-scattered waves,” Nat. Photonics 9, 253–258 (2015).

S. Schott, J. Bertolotti, J.-F. Léger, L. Bourdieu, and S. Gigan, “Characterization of the angular memory effect of scattered light in biological tissues,” Opt. Exp. 23, 13505–13516 (2015).
[Crossref]

2014 (1)

O. Assayag, M. Antoine, B. Sigal-Zafrani, M. Riben, F. Harms, A. Burcheri, B. Le Conte de Poly, and A. C. Boccara, “Large field, high resolution full field optical coherence tomography: A pre-clinical study of human breast tissue and cancer assessment,” Technol. Cancer Res. Treat. 13, 455–468 (2014).

2013 (2)

2012 (1)

N. Ji, T. Sato, and E. Betzig, “Characterization and adaptive optical correction of aberrations during in vivo imaging in the mouse cortex,” Proc. Natl. Acad. Sci. USA 109, 22–27 (2012).
[Crossref]

2011 (1)

S. M. Popoff, A. Aubry, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Exploiting the time-reversal operator for adaptive optics, selective focusing, and scattering pattern analysis,” Phys. Rev. Lett. 107, 263901 (2011).
[Crossref]

2010 (3)

V. Ntziachristos, “Going deeper than microscopy: the optical imaging frontier in biology,” Nat. Methods 7, 603–614 (2010).
[Crossref] [PubMed]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[Crossref] [PubMed]

S. M. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 81 (2010).
[Crossref] [PubMed]

2009 (2)

A. Aubry and A. Derode, “Random matrix theory applied to acoustic backscattering and imaging In complex media,” Phys. Rev. Lett. 102, 084301 (2009).
[Crossref] [PubMed]

D. L. Marks, B. J. Davis, S. A. Boppart, and P. S. Carney, “Partially coherent illumination in full-field interferometric synthetic aperture microscopy,” J. Opt. Soc. Am. A 26, 376–386 (2009).
[Crossref]

2007 (1)

2005 (2)

2004 (1)

2003 (1)

C. Dunsby and P. M. W. French, “Techniques for depth-resolved imaging through turbid media including coherence-gated imaging,” J. Phys. D: Appl. Phys. 248, R207–R227 (2003).
[Crossref]

2002 (2)

L. Vabre, A. Dubois, and A. C. Boccara, “Thermal-light full-field optical coherence tomography”, Opt. Lett. 27, 130–132 (2002).
[Crossref]

A. Dubois, L. Vabre, A. C. Boccara, and E. Beaurepaire, “High resolution full-field optical coherence tomography with a Linnik microscope,” Appl. Lett. 41, 805–812 (2002).

1999 (2)

K. K. Bizheva, A. M. Siegel, and D. A. Boas, “Path-length-resolved dynamic light scattering in highly scattering random media: The transition to diffusing wave spectroscopy,” Phys. Rev. E 58, 7664–7767 (1999).
[Crossref]

G. Yao and L. V. Wang, “Monte Carlo simulation of an optical coherence tomography signal in homogeneous turbid media,” Phys. Med. Biol. 44, 2307–2320 (1999).
[Crossref] [PubMed]

1997 (2)

1996 (1)

1995 (1)

1994 (2)

J. M. Schmitt, A. Knüttel, and M. J. Yadlowsky, “Confocal microscopy in turbid media,” J. Opt. Soc. Am. A 11, 2226–2235 (1994).
[Crossref]

C. Prada and M. Fink, “Eigenmodes of the time reversal operator: A solution to selective focusing in multiple-target media,” Wave Motion 20, 151–163 (1994).
[Crossref]

1991 (3)

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

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

J. Xhu, D. Pine, and D. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
[Crossref]

1990 (1)

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[Crossref] [PubMed]

1988 (1)

E. Akkermans, P. E. Wolf, R. Maynard, and G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. France 49, 77–98 (1988).
[Crossref]

1987 (1)

A. Z. Genack, “Optical transmission in disordered media,” Phys. Rev. Lett. 58, 2043–2046 (1987).
[Crossref] [PubMed]

1986 (1)

1985 (2)

M. P. V. Albada and A. Lagendijk, “Observation of weak localization of light in a random medium,” Phys. Rev. Lett. 55, 2692–2695 (1985).
[Crossref] [PubMed]

P.-E. Wolf and G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
[Crossref] [PubMed]

1982 (1)

1972 (1)

J. T. Edwards and D. J. Thouless, “Numerical studies of localization in disordered systems,” J. Phys. C: Solid State Phys. 5, 807 (1972).
[Crossref]

Akkermans, E.

E. Akkermans, P. E. Wolf, R. Maynard, and G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. France 49, 77–98 (1988).
[Crossref]

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

Albada, M. P. V.

M. P. V. Albada and A. Lagendijk, “Observation of weak localization of light in a random medium,” Phys. Rev. Lett. 55, 2692–2695 (1985).
[Crossref] [PubMed]

Alfano, R. R.

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

Andersen, P.

P. Andersen, T. M. Jørgensen, L. Thrane, A. Tycho, and H. T. Yura, “Modeling Light-Tissue Interaction in Optical Coherence Tomography Systems,” in Optical Coherence Tomography: Technology and Applications (Springer-Verlag, 2008).
[Crossref]

Andersen, P. E.

Antoine, M.

O. Assayag, M. Antoine, B. Sigal-Zafrani, M. Riben, F. Harms, A. Burcheri, B. Le Conte de Poly, and A. C. Boccara, “Large field, high resolution full field optical coherence tomography: A pre-clinical study of human breast tissue and cancer assessment,” Technol. Cancer Res. Treat. 13, 455–468 (2014).

Apelian, C.

Assayag, O.

O. Assayag, M. Antoine, B. Sigal-Zafrani, M. Riben, F. Harms, A. Burcheri, B. Le Conte de Poly, and A. C. Boccara, “Large field, high resolution full field optical coherence tomography: A pre-clinical study of human breast tissue and cancer assessment,” Technol. Cancer Res. Treat. 13, 455–468 (2014).

Aubry, A.

A. Badon, D. Li, G. Lerosey, A. C. Boccara, M. Fink, and A. Aubry, “Smart optical coherence tomography for ultra-deep imaging through highly scattering media,” Sci. Adv. 2, e1600370 (2016).
[Crossref] [PubMed]

S. M. Popoff, A. Aubry, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Exploiting the time-reversal operator for adaptive optics, selective focusing, and scattering pattern analysis,” Phys. Rev. Lett. 107, 263901 (2011).
[Crossref]

A. Aubry and A. Derode, “Random matrix theory applied to acoustic backscattering and imaging In complex media,” Phys. Rev. Lett. 102, 084301 (2009).
[Crossref] [PubMed]

Badon, A.

A. Badon, D. Li, G. Lerosey, A. C. Boccara, M. Fink, and A. Aubry, “Smart optical coherence tomography for ultra-deep imaging through highly scattering media,” Sci. Adv. 2, e1600370 (2016).
[Crossref] [PubMed]

Beaurepaire, E.

A. Dubois, L. Vabre, A. C. Boccara, and E. Beaurepaire, “High resolution full-field optical coherence tomography with a Linnik microscope,” Appl. Lett. 41, 805–812 (2002).

Bertolotti, J.

S. Schott, J. Bertolotti, J.-F. Léger, L. Bourdieu, and S. Gigan, “Characterization of the angular memory effect of scattered light in biological tissues,” Opt. Exp. 23, 13505–13516 (2015).
[Crossref]

Betzig, E.

N. Ji, T. Sato, and E. Betzig, “Characterization and adaptive optical correction of aberrations during in vivo imaging in the mouse cortex,” Proc. Natl. Acad. Sci. USA 109, 22–27 (2012).
[Crossref]

Bizheva, K. K.

K. K. Bizheva, A. M. Siegel, and D. A. Boas, “Path-length-resolved dynamic light scattering in highly scattering random media: The transition to diffusing wave spectroscopy,” Phys. Rev. E 58, 7664–7767 (1999).
[Crossref]

Boas, D. A.

K. K. Bizheva, A. M. Siegel, and D. A. Boas, “Path-length-resolved dynamic light scattering in highly scattering random media: The transition to diffusing wave spectroscopy,” Phys. Rev. E 58, 7664–7767 (1999).
[Crossref]

Boccara, A. C.

O. Thouvenin, K. Grieve, P. Xiao, C. Apelian, and A. C. Boccara, “En face coherence microscopy,” Biomed. Opt. Express 8, 622–639 (2017).
[Crossref] [PubMed]

A. Badon, D. Li, G. Lerosey, A. C. Boccara, M. Fink, and A. Aubry, “Smart optical coherence tomography for ultra-deep imaging through highly scattering media,” Sci. Adv. 2, e1600370 (2016).
[Crossref] [PubMed]

O. Assayag, M. Antoine, B. Sigal-Zafrani, M. Riben, F. Harms, A. Burcheri, B. Le Conte de Poly, and A. C. Boccara, “Large field, high resolution full field optical coherence tomography: A pre-clinical study of human breast tissue and cancer assessment,” Technol. Cancer Res. Treat. 13, 455–468 (2014).

S. M. Popoff, A. Aubry, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Exploiting the time-reversal operator for adaptive optics, selective focusing, and scattering pattern analysis,” Phys. Rev. Lett. 107, 263901 (2011).
[Crossref]

S. M. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 81 (2010).
[Crossref] [PubMed]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[Crossref] [PubMed]

A. Dubois, L. Vabre, A. C. Boccara, and E. Beaurepaire, “High resolution full-field optical coherence tomography with a Linnik microscope,” Appl. Lett. 41, 805–812 (2002).

L. Vabre, A. Dubois, and A. C. Boccara, “Thermal-light full-field optical coherence tomography”, Opt. Lett. 27, 130–132 (2002).
[Crossref]

Bonner, R. F.

Boppart, S. A.

Born, M.

M. Born and E. Wolf, Principle of Optics, 7th ed (Cambridge University, 1999).
[Crossref]

Bourdieu, L.

S. Schott, J. Bertolotti, J.-F. Léger, L. Bourdieu, and S. Gigan, “Characterization of the angular memory effect of scattered light in biological tissues,” Opt. Exp. 23, 13505–13516 (2015).
[Crossref]

Bourquin, S.

Burcheri, A.

O. Assayag, M. Antoine, B. Sigal-Zafrani, M. Riben, F. Harms, A. Burcheri, B. Le Conte de Poly, and A. C. Boccara, “Large field, high resolution full field optical coherence tomography: A pre-clinical study of human breast tissue and cancer assessment,” Technol. Cancer Res. Treat. 13, 455–468 (2014).

Carminati, R.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[Crossref] [PubMed]

Carney, P. S.

Chang, W.

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

Choi, H.

Choi, W.

S. Kang, S. Jeong, W. Choi, H. Ko, T. D. Yang, J. H. Joo, J.-S. Lee, Y.-S. Lim, Q. H. Park, and W. Choi, “Imaging deep within a scattering medium using collective accumulation of single-scattered waves,” Nat. Photonics 9, 253–258 (2015).

S. Kang, S. Jeong, W. Choi, H. Ko, T. D. Yang, J. H. Joo, J.-S. Lee, Y.-S. Lim, Q. H. Park, and W. Choi, “Imaging deep within a scattering medium using collective accumulation of single-scattered waves,” Nat. Photonics 9, 253–258 (2015).

Choma, M. A.

J. A. Izatt and M. A. Choma, “Theory of optical coherence tomography,” in Optical Coherence Tomography: Technology and Applications, W. Drexler and J. G. Fujimoto, eds. (Springer-Verlag, 2008).
[Crossref]

Cui, M.

J. H. Park, W. Sun, and M. Cui, “High-resolution in vivo imaging of mouse brain through the intact skull,” Proc. Natl. Acad. Sci. U. S. A. 112, 9236–9241 (2015).
[Crossref] [PubMed]

Davis, B. J.

De Silvestri, S.

Denk, W.

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[Crossref] [PubMed]

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Appl. Lett. (1)

A. Dubois, L. Vabre, A. C. Boccara, and E. Beaurepaire, “High resolution full-field optical coherence tomography with a Linnik microscope,” Appl. Lett. 41, 805–812 (2002).

Appl. Opt. (1)

Biomed. Opt. Express (1)

J. Opt. Soc. Am. (1)

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

J. Phys. C: Solid State Phys. (1)

J. T. Edwards and D. J. Thouless, “Numerical studies of localization in disordered systems,” J. Phys. C: Solid State Phys. 5, 807 (1972).
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J. Phys. D: Appl. Phys. (1)

C. Dunsby and P. M. W. French, “Techniques for depth-resolved imaging through turbid media including coherence-gated imaging,” J. Phys. D: Appl. Phys. 248, R207–R227 (2003).
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J. Phys. France (1)

E. Akkermans, P. E. Wolf, R. Maynard, and G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. France 49, 77–98 (1988).
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Nat. Commun. (1)

S. M. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 81 (2010).
[Crossref] [PubMed]

Nat. Methods (1)

V. Ntziachristos, “Going deeper than microscopy: the optical imaging frontier in biology,” Nat. Methods 7, 603–614 (2010).
[Crossref] [PubMed]

Nat. Photonics (1)

S. Kang, S. Jeong, W. Choi, H. Ko, T. D. Yang, J. H. Joo, J.-S. Lee, Y.-S. Lim, Q. H. Park, and W. Choi, “Imaging deep within a scattering medium using collective accumulation of single-scattered waves,” Nat. Photonics 9, 253–258 (2015).

Opt. Exp. (1)

S. Schott, J. Bertolotti, J.-F. Léger, L. Bourdieu, and S. Gigan, “Characterization of the angular memory effect of scattered light in biological tissues,” Opt. Exp. 23, 13505–13516 (2015).
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Opt. Express (1)

Opt. Lett. (4)

Phys. Med. Biol. (2)

S. Jacques, “Optical properties of biological tissues: a review,” Phys. Med. Biol. 58, R37–R61 (2013).
[Crossref] [PubMed]

G. Yao and L. V. Wang, “Monte Carlo simulation of an optical coherence tomography signal in homogeneous turbid media,” Phys. Med. Biol. 44, 2307–2320 (1999).
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Phys. Rev. A (1)

J. Xhu, D. Pine, and D. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
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Phys. Rev. E (1)

K. K. Bizheva, A. M. Siegel, and D. A. Boas, “Path-length-resolved dynamic light scattering in highly scattering random media: The transition to diffusing wave spectroscopy,” Phys. Rev. E 58, 7664–7767 (1999).
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Figures (7)

Fig. 1
Fig. 1

Imaging configuration. (a) A specimen in the focal plane of a MO is illuminated by a source at position ri in the back focal plane of a lens Li. The reflected light is collected through the same MO and measured in the focal plane of a lens Lo. (b) Plane wave illumination configuration. (c) Focused illumination configuration.

Fig. 2
Fig. 2

(a) Imaging of a target embedded at depth F inside a scattering medium of scattering mean-free path s. Both the incident and reflected wave-fields can be scattered by the heterogeneities of the medium before or after reaching the target. (b) Between the first and the last scattering event, the propagation of light is modeled with a propagator Γ using the radiative transfer or diffusion theories.

Fig. 3
Fig. 3

(a) Single-to-multiple scattering ratio as a function of the optical depth F/ℓs for a point-like target. The performances of conventional microscopy (black line), confocal microscopy (blue line), OCT (red line) and smart OCT (green line) are compared. The y-axis is in log-scale. These curves have been computed considering experimental parameters typical of full-field OCT (see Table 1). The detection threshold (SMR ∼ 1, black dashed horizontal line) yields an imaging depth limit of ∼ 4s for confocal microscopy, ∼ 9s for OCT and ∼ 17s for smart-OCT. (b) Evolution of the corresponding penetration depths derived for each imaging technique as a function of the NA of the MO. In both (a) and (b), the SMRs in OCT and smart-OCT have been computed by considering either the diffusive approximation [Eq. (39)] or a Monte-Carlo calculation of the time-dependent albedo (continuous and dashed lines, respectively).

Fig. 4
Fig. 4

(a) Time-dependent impulse response for a point-like target embedded in a scattering medium. This response can be expressed as the sum of the MS background (in blue) and the target echo arriving at time t = tB (in red). The time-of-flight distribution of MS photons can be either computed by means of a Monte-Carlo simulation (continuous line) or analytically predicted by the diffusion theory at large optical depths (dashed line, Eq. (39)). Coherence time gating allows to select the photons over a time window centered around this ballistic time of flight and thus drastically enhance the SMR. (b) The corresponding isochronous volume is a slab of thickness cω centered around the focal plane.

Fig. 5
Fig. 5

(a) Scheme of an extended target placed behind a scattering layer of thickness L. (b) Single-to-multiple scattering ratio as a function of the optical thickness L/ℓs for an extended target. The performances of conventional microscopy (black line), confocal microscopy (blue line) and OCT (red line) are compared. The y-axis is in log-scale. These curves have been computed considering experimental parameters typical of full-field OCT (see Table 1) and for a target reflectivity |ρ| = 1. The detection threshold (SMR∼ 1, black dashed horizontal line) yields an imaging depth limit of ∼ 3.5s for conventional microscopy, ∼ 7.5s for confocal microscopy, ∼ 12s for OCT. (c) Evolution of the corresponding penetration depths derived for each imaging technique as a function of the NA of the MO. In panels (a) and (b), the SMR in OCT has been computed by considering either the diffusive approximation [Eq. (39)] or a Monte-Carlo calculation of the time-dependent albedo (continuous and dashed lines, respectively).

Fig. 6
Fig. 6

Modeling of soft tissues. The biological media are made of randomly distributed scatterers at a concentration n. The brightness of tissues can be quantified by the back-scattering cross-section σb[Eq. (55)].

Fig. 7
Fig. 7

Imaging depth limit in biological tissues considering experimental parameters typical of full-field OCT (see Table 1). (a) Single-to-multiple scattering ratio as a function of the optical thickness F/ℓs. The performances of confocal microscopy (blue line) and OCT (red line) are compared. The SMR in OCT has been computed by considering either the diffusive approximation [Eq. (39)] or a Monte-Carlo calculation of the time-dependent albedo (continuous and dashed lines, respectively). (b) Evolution of the penetration depths derived for confocal microscopy (blue continuous line) and OCT (diffusive prediction, dashed line) as a function of the anisotropy factor η.

Tables (1)

Tables Icon

Table 1 Experimental parameters used for the theoretical prediction of the single-to-multiple scattering ratio.

Equations (61)

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g S ( u o , u i , ω ) = d 3 r g l ( u o , r , ω ) v ( r ) g l ( r , u i , ω )
g l ( r , u i , ω ) = e F z 2 μ i s e i k ( 2 f z ) e i k z 2 f 2 | r | 2 e i k f r u i i λ f p ( u i )
| p ( u i ) | = { 1 inside the lens aperture 0 otherwise
v ( r ) = σ δ ( r )
g S ( u o , u i , ω ) = σ ( λ f ) 2 e F 2 s ( 1 μ i + 1 μ o ) e 4 i k f p ( u o ) p ( u i ) .
| g M ( u o , u i , ω ) | 2 Ω d r 1 Ω d r 2 | g l ( u o , r 2 , ω ) | 2 Γ ( r 2 , r 1 ) | g l ( r 1 , u i , ω ) | 2
| g M ( u o , u i , ω ) | 2 = p ( u o ) p ( u i ) ( λ f ) 4 Ω d r 1 Ω d r 2 exp [ ( z 1 + z 2 ) / s ] Γ ( r 2 , r 1 ) .
α = 1 ( 4 π ) 2 W 2 Ω d r 1 Ω d r 2 exp [ ( z 1 + z 2 ) / s ] Γ ( r 2 , r 1 ) .
α ( μ r ) ~ 3 4 π μ r ( μ r + z 0 t ) ( 1 5 3 t L + 2 z 0 )
z 0 = 2 3 t 1 + R 1 R
α 1 2 sin 2 ( β / 2 ) cos β 1 d μ r α ( μ r )
| g M ( u o , u i , ω ) | 2 = ( 4 π ) 2 α W 2 ( λ f ) 4 p ( u o ) p ( u i ) .
l c ~ λ f W .
SMR p = 1 ( 4 π ) 2 σ W e F S ( 1 μ i + 1 μ O ) α .
g ( r o , u i , ω ) = e 2 i k f i λ f d 2 u o g ( u o , u i , ω ) e i k f u o . r o .
g S ( r o , u i , ω ) = i e 6 i k f σ e F / ( 2 s μ i ) e F / ( 2 s ) δ r 2 ( λ f ) h ( r o ) .
e F / ( 2 s ) = 1 2 sin 2 ( β / 2 ) cos β 1 d μ e F / ( 2 s μ ) .
h ( r o ) = 1 A d 2 u o p ( u o ) e i k f u o . r o
A = ( λ f δ r ) 2
W 2 | g M ( r o , u i , ω ) | 2 = A | g M ( u o , u i , ω ) | 2
| g M ( r o , u i , ω ) | 2 = ( 4 π ) 2 α δ r 2 ( λ f ) 2 p ( u i ) .
SMR m = S ( 4 π ) 2 σ δ r 2 e 2 F / s α
g ( r o , u i , ω ) = e 2 i k f i λ f d u i g ( r o , u i , ω ) e i 2 π λ f u i . r i .
g S ( r o , u i , ω ) = e 8 i k f σ e F / s δ r 4 h ( r o ) h ( r i ) .
W 2 | g M ( r o , u i , ω ) | 2 = A | g M ( r o , u i , ω ) | 2 .
| g M ( r o , u i , ω ) | 2 = ( 4 π ) 2 δ r 4 W 2 α .
SMR c = S 2 ( 4 π ) 2 σ δ r 2 ( W δ r ) 2 e 2 F / s α .
f ( r o , r i , t ) = ψ ( r o , r i , t ) * s ( t ) .
f ( r o , r i , t ) = g ( r o , r i , t ) * s ( t ) * s ( t )
f ( r o , r i , t ) = d ω g ( r o , r i , ω ) | s ( ω ) | 2 e i ω t .
f S ( r o , r i , t ) = d ω g S ( r o , r i , t ) | s ( ω ) | 2 e i ω t .
f S ( r o , r i , t B ) ~ σ δ r 4 e F / s Δ ω | s ( ω 0 ) | 2 h ( r o ) h ( r i ) .
| f M ( r o , r i , t ) | 2 = d ω 1 d ω 2 g M ( r o , r i , ω 1 ) g M * ( r o , r i , ω 2 ) | s ( ω 1 ) | 2 | s ( ω 2 ) | 2 e i ( ω 2 ω 1 ) t
C ( r o , r i , ω 2 ω 1 ) = g M ( r o , r i , ω 1 ) g M * ( r o , r i , ω 2 ) .
| f M ( r o , r i , t ) | 2 = Δ ω | s 0 ( ω ) | 4 d ω C ( r o , r i , ω ) e i ω t .
| f M ( r o , r i , t ) | 2 ~ Δ ω | s 0 ( ω 0 ) | 4 | g M ( r o , r i , t ) | 2 .
| g M ( r o , r i , t ) | 2 = ( 4 π ) 2 α ( t ) δ r 4 W 2
α ( t ) = 1 ( 4 π ) 2 W 2 d r 1 d r 2 e ( z 1 + z 2 ) / s Γ ( r 2 , r 1 , t ) .
α ( t ) c ( z 0 + s ) 2 ( 4 π D t ) 3 / 2 .
| f M ( r o , r i , t ) | 2 ~ ( 4 π ) 2 α ( t ) δ r 4 W 2 Δ ω | s 0 ( ω 0 ) | 4 .
SMR t = S 2 ( 4 π ) 2 σ δ r 2 ( W δ r ) 2 e 2 F / s Δ ω α ( t B ) .
g S ( u o , u i , ω ) = 2 ρ d 2 r g l ( u o , r , ω ) n g l ( r , u i , ω ) .
g S ( u o , u i , ω ) = 2 i e 4 i k f k ρ e L / ( s μ o ) δ r ( u i + u o )
g S ( r o , u i , ω ) = e 6 i k f 2 k ρ λ f e 2 L / s p ( u i ) e i k f u i . r o
e L / s = 1 2 sin 2 ( β / 2 ) cos β 1 d μ o e L / ( s μ o ) .
SMR m = ( | ρ | 2 N A ) 2 e 2 L / s α .
g S ( r o , u i , ω ) = e 8 i k f 2 k ρ e L / s δ r 2 h ( r o r i ) .
SMR c = S | ρ | 2 ( W λ ) 2 e 2 L / s α .
f S ( r o , r i , t B ) ~ 2 ρ k δ r 2 e L / s Δ ω | s 0 ( ω 0 ) | 2 h ( r o r i )
SMR t = S | ρ | 2 ( W λ ) 2 e 2 L / s Δ ω α ( t B )
σ = 4 π n d σ s d Ω V = n σ s V
σ = V s
σ b = 4 π n d σ d Ω N A V
d σ d Ω N A = 1 2 sin 2 ( β / 2 ) 0 β d θ d σ ( θ ) d Ω sin θ
σ b = V b .
p = 2 n λ / ( N A ) 2
σ b = p δ r 2 b .
SMR m = S ( 4 π ) 2 p b e 2 F / s α .
SMR c = S 2 ( 4 π ) 2 p b ( W δ r ) 2 e 2 F / s α = n 3 S 2 2 π 2 W 2 λ b e 2 F / s α .
σ = c δ r 2 2 b Δ ω .
SMR t = S 2 ( 4 π ) 2 c 2 b ( W δ r ) 2 e 2 F / s α ( t B ) .

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