Abstract

Diffuse photon density waves have lately been used both to characterize diffusive media and to locate and characterize hidden objects, such as tumors, in soft tissue. In practice, most biological media of medical interest consist of various layers with different optical properties, such as the fat layer in the breast or the different layers present in the skin. Also, most experimental setups consist of a multilayered system, where the medium to be characterized (i.e., the patient’s organ) is usually bounded by optically diffusive plates. Incorrect modeling of interfaces may induce errors comparable to the weak signals obtained from tumors embedded deep in highly heterogeneous tissue and lead to significant reconstruction artifacts. To provide a means to analyze the data acquired in these configurations, the basic expressions for the reflection and transmission coefficients for diffusive–diffusive and diffusive–nondiffusive interfaces are presented. A comparison is made between a diffusive slab and an ordinary dielectric slab, thus establishing the limiting distance between the two interfaces of the slab for multiple reflections between them to be considered important. A rigorous formulation for multiple-layered (M-layered) diffusive media is put forward, and a method for solving any M-layered medium is shown. The theory presented is used to characterize a two-layered medium from transmission measurements, showing that the coefficients of scattering, μs, and absorption, μa, are retrieved with great accuracy. Finally, we demonstrate the simultaneous retrieval of both μs and μa.

© 2001 Optical Society of America

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2000 (3)

1999 (9)

J. Ripoll, M. Nieto-Vesperinas, “Reflection and transmission coefficients for diffuse photon-density waves,” Opt. Lett. 24, 796–798 (1999).
[CrossRef]

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

L. O. Svaasand, T. Spott, J. B. Frishkin, T. Pham, B. J. Tromberg, M. W. Berns, “Reflectance measurements of layered media with diffuse photon-density waves: a potential tool for evaluating deep burns and subcutaneous lesions,” Phys. Med. Biol. 44, 801–813 (1999).
[CrossRef] [PubMed]

T. Durduran, J. P. Culver, M. J. Holboke, X. D. Li, L. Zubkov, B. Chance, D. N. Pattanayak, A. G. Yodh, “Algorithms for 3D localization and imaging using near-field diffraction tomography with diffuse light,” Opt. Exp. 4, 247–262 (1999).
[CrossRef]

D. N. Pattanayak, A. G. Yodh, “Diffuse optical 3D-slice imaging of bounded turbid media using a new integrodifferential equation,” Opt. Exp. 4, 231–240 (1999).
[CrossRef]

V. Ntziachristos, X. Ma, A. G. Yodh, B. Chance, “Multichannel photon counting instrument for spatially resolved near-infrared spectroscopy,” Rev. Sci. Instrum. 70, 193–201 (1999).
[CrossRef]

V. Ntziachristos, B. Chance, A. G. Yodh, “Differential diffuse optical tomography,” Opt. Exp. 5, 230–242 (1999).
[CrossRef]

J. Ripoll, M. Nieto-Vesperinas, R. Carminati, “Spatial resolution of diffuse photon density waves,” J. Opt. Soc. Am. A 16, 1466–1476 (1999).
[CrossRef]

J. Ripoll, M. Nieto-Vesperinas, “Index mismatch for diffuse photon-density waves both at flat and rough diffuse-diffuse interfaces,” J. Opt. Soc. Am. A 16, 1947–1957 (1999).
[CrossRef]

1998 (6)

1997 (1)

1996 (2)

E. B. de Haller, “Time-resolved transillumination and optical tomography,” J. Biomed. Opt. 1, 7–17 (1996).
[CrossRef] [PubMed]

A. H. Hielscher, H. Liu, B. Chance, “Time-resolved photon emission from layered turbid media,” Appl. Opt. 35, 2221–2227 (1996).
[CrossRef]

1995 (3)

1994 (1)

1993 (1)

1992 (2)

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef] [PubMed]

I. Dayan, S. Havlin, G. H. Weiss, “Photon migration in a two-layer turbid medium: a diffusion analysis,” J. Mod. Opt. 39, 1567–1582 (1992).
[CrossRef]

1990 (1)

1989 (1)

1988 (1)

Alexandrakis, G.

Aronson, R.

Arridge, S. R.

J. Ripoll, S. R. Arridge, H. Dehghani, M. Nieto-Vesperinas, “Boundary conditions for light propagation in diffusive media with nonscattering regions,” J. Opt. Soc. Am. A 17, 1671–1681 (2000).
[CrossRef]

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, “Reconstruction methods for near-infrared absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 204–215 (1991).
[CrossRef]

Banos, A.

A. Banos, Dipole Radiation in the Presence of a Conducting Half-Space (Pergamon, New York, 1966).

Berns, M. W.

L. O. Svaasand, T. Spott, J. B. Frishkin, T. Pham, B. J. Tromberg, M. W. Berns, “Reflectance measurements of layered media with diffuse photon-density waves: a potential tool for evaluating deep burns and subcutaneous lesions,” Phys. Med. Biol. 44, 801–813 (1999).
[CrossRef] [PubMed]

Boas, D. A.

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef] [PubMed]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1993).

Carminati, R.

Chance, B.

V. Ntziachristos, X. Ma, A. G. Yodh, B. Chance, “Multichannel photon counting instrument for spatially resolved near-infrared spectroscopy,” Rev. Sci. Instrum. 70, 193–201 (1999).
[CrossRef]

V. Ntziachristos, B. Chance, A. G. Yodh, “Differential diffuse optical tomography,” Opt. Exp. 5, 230–242 (1999).
[CrossRef]

T. Durduran, J. P. Culver, M. J. Holboke, X. D. Li, L. Zubkov, B. Chance, D. N. Pattanayak, A. G. Yodh, “Algorithms for 3D localization and imaging using near-field diffraction tomography with diffuse light,” Opt. Exp. 4, 247–262 (1999).
[CrossRef]

V. Ntziachristos, X. Ma, B. Chance, “Time-correlated single photon counting imager for simultaneous magnetic resonance and near-infrared mammography,” Rev. Sci. Instrum. 69, 4221–4233 (1998).
[CrossRef]

X. De Li, T. Durduran, A. G. Yodh, B. Chance, D. N. Pattanayak, “Diffraction tomography for biochemical imaging with diffuse-photon density waves,” Opt. Lett. 22, 573–575 (1997).
[CrossRef]

A. H. Hielscher, H. Liu, B. Chance, “Time-resolved photon emission from layered turbid media,” Appl. Opt. 35, 2221–2227 (1996).
[CrossRef]

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 38–40 (1995).
[CrossRef]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef] [PubMed]

M. S. Patterson, B. Chance, B. C. Wilson, “Time-resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
[CrossRef] [PubMed]

Cope, M.

S. R. Arridge, P. Van Der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for near-infrared absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 204–215 (1991).
[CrossRef]

Culver, J. P.

T. Durduran, J. P. Culver, M. J. Holboke, X. D. Li, L. Zubkov, B. Chance, D. N. Pattanayak, A. G. Yodh, “Algorithms for 3D localization and imaging using near-field diffraction tomography with diffuse light,” Opt. Exp. 4, 247–262 (1999).
[CrossRef]

Dayan, I.

I. Dayan, S. Havlin, G. H. Weiss, “Photon migration in a two-layer turbid medium: a diffusion analysis,” J. Mod. Opt. 39, 1567–1582 (1992).
[CrossRef]

de Haller, E. B.

E. B. de Haller, “Time-resolved transillumination and optical tomography,” J. Biomed. Opt. 1, 7–17 (1996).
[CrossRef] [PubMed]

De Li, X.

Dehghani, H.

Delpy, D. T.

S. R. Arridge, P. Van Der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for near-infrared absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 204–215 (1991).
[CrossRef]

Dögnitz, N.

Durduran, T.

T. Durduran, J. P. Culver, M. J. Holboke, X. D. Li, L. Zubkov, B. Chance, D. N. Pattanayak, A. G. Yodh, “Algorithms for 3D localization and imaging using near-field diffraction tomography with diffuse light,” Opt. Exp. 4, 247–262 (1999).
[CrossRef]

X. De Li, T. Durduran, A. G. Yodh, B. Chance, D. N. Pattanayak, “Diffraction tomography for biochemical imaging with diffuse-photon density waves,” Opt. Lett. 22, 573–575 (1997).
[CrossRef]

Essenpreis, M.

Fantini, S.

Farrell, T. J.

Feng, T.

Franceschini, M. A.

Frishkin, J. B.

L. O. Svaasand, T. Spott, J. B. Frishkin, T. Pham, B. J. Tromberg, M. W. Berns, “Reflectance measurements of layered media with diffuse photon-density waves: a potential tool for evaluating deep burns and subcutaneous lesions,” Phys. Med. Biol. 44, 801–813 (1999).
[CrossRef] [PubMed]

J. B. Frishkin, E. Gratton, “Propagation of photon-density waves in strongly scattering media containing an absorbing semi-infinite plane bounded by a straight edge,” J. Opt. Soc. Am. A 10, 127–140 (1993).
[CrossRef]

Glanzmann, T.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Gratton, E.

Haskell, R. C.

Havlin, S.

I. Dayan, S. Havlin, G. H. Weiss, “Photon migration in a two-layer turbid medium: a diffusion analysis,” J. Mod. Opt. 39, 1567–1582 (1992).
[CrossRef]

Hielscher, A. H.

Holboke, M. J.

T. Durduran, J. P. Culver, M. J. Holboke, X. D. Li, L. Zubkov, B. Chance, D. N. Pattanayak, A. G. Yodh, “Algorithms for 3D localization and imaging using near-field diffraction tomography with diffuse light,” Opt. Exp. 4, 247–262 (1999).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1.

Jiang, H.

Kaschke, M.

Keijzer, M.

Kienle, A.

Li, X. D.

T. Durduran, J. P. Culver, M. J. Holboke, X. D. Li, L. Zubkov, B. Chance, D. N. Pattanayak, A. G. Yodh, “Algorithms for 3D localization and imaging using near-field diffraction tomography with diffuse light,” Opt. Exp. 4, 247–262 (1999).
[CrossRef]

Liu, H.

Ma, X.

V. Ntziachristos, X. Ma, A. G. Yodh, B. Chance, “Multichannel photon counting instrument for spatially resolved near-infrared spectroscopy,” Rev. Sci. Instrum. 70, 193–201 (1999).
[CrossRef]

V. Ntziachristos, X. Ma, B. Chance, “Time-correlated single photon counting imager for simultaneous magnetic resonance and near-infrared mammography,” Rev. Sci. Instrum. 69, 4221–4233 (1998).
[CrossRef]

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge UK, 1995).

McAdams, M. S.

Moesta, K. T.

Nieto-Vesperinas, M.

Ntziachristos, V.

V. Ntziachristos, X. Ma, A. G. Yodh, B. Chance, “Multichannel photon counting instrument for spatially resolved near-infrared spectroscopy,” Rev. Sci. Instrum. 70, 193–201 (1999).
[CrossRef]

V. Ntziachristos, B. Chance, A. G. Yodh, “Differential diffuse optical tomography,” Opt. Exp. 5, 230–242 (1999).
[CrossRef]

V. Ntziachristos, X. Ma, B. Chance, “Time-correlated single photon counting imager for simultaneous magnetic resonance and near-infrared mammography,” Rev. Sci. Instrum. 69, 4221–4233 (1998).
[CrossRef]

O’Leary, M. A.

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef] [PubMed]

Osterberg, U. L.

Pattanayak, D. N.

D. N. Pattanayak, A. G. Yodh, “Diffuse optical 3D-slice imaging of bounded turbid media using a new integrodifferential equation,” Opt. Exp. 4, 231–240 (1999).
[CrossRef]

T. Durduran, J. P. Culver, M. J. Holboke, X. D. Li, L. Zubkov, B. Chance, D. N. Pattanayak, A. G. Yodh, “Algorithms for 3D localization and imaging using near-field diffraction tomography with diffuse light,” Opt. Exp. 4, 247–262 (1999).
[CrossRef]

X. De Li, T. Durduran, A. G. Yodh, B. Chance, D. N. Pattanayak, “Diffraction tomography for biochemical imaging with diffuse-photon density waves,” Opt. Lett. 22, 573–575 (1997).
[CrossRef]

Patterson, M. S.

Paulsen, K. D.

Pham, T.

L. O. Svaasand, T. Spott, J. B. Frishkin, T. Pham, B. J. Tromberg, M. W. Berns, “Reflectance measurements of layered media with diffuse photon-density waves: a potential tool for evaluating deep burns and subcutaneous lesions,” Phys. Med. Biol. 44, 801–813 (1999).
[CrossRef] [PubMed]

Pham, T. H.

Pogue, B. W.

Ripoll, J.

Schlag, P. M.

Schmitt, J. M.

Spott, T.

T. H. Pham, T. Spott, L. O. Svaasand, B. J. Tromberg, “Quantifying the properties of two-layer turbid media with frequency-domain diffuse reflectance,” Appl. Opt. 39, 4733–4745 (2000).
[CrossRef]

L. O. Svaasand, T. Spott, J. B. Frishkin, T. Pham, B. J. Tromberg, M. W. Berns, “Reflectance measurements of layered media with diffuse photon-density waves: a potential tool for evaluating deep burns and subcutaneous lesions,” Phys. Med. Biol. 44, 801–813 (1999).
[CrossRef] [PubMed]

Star, W. M.

Storchi, P. R. M.

Svaasand, L. O.

Tromberg, B. J.

Tsay, T.

Van Der Zee, P.

S. R. Arridge, P. Van Der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for near-infrared absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 204–215 (1991).
[CrossRef]

Wagnieres, G.

Walker, E. C.

Walker, S. A.

Weiss, G. H.

I. Dayan, S. Havlin, G. H. Weiss, “Photon migration in a two-layer turbid medium: a diffusion analysis,” J. Mod. Opt. 39, 1567–1582 (1992).
[CrossRef]

Wilson, B. C.

Wolf, E.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge UK, 1995).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1993).

Yariv, A.

A. Yariv, Introduction to Optical Electronics, 2nd ed. (Holt, Rinehart & Winston, New York, 1976).

Yodh, A.

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 38–40 (1995).
[CrossRef]

Yodh, A. G.

T. Durduran, J. P. Culver, M. J. Holboke, X. D. Li, L. Zubkov, B. Chance, D. N. Pattanayak, A. G. Yodh, “Algorithms for 3D localization and imaging using near-field diffraction tomography with diffuse light,” Opt. Exp. 4, 247–262 (1999).
[CrossRef]

D. N. Pattanayak, A. G. Yodh, “Diffuse optical 3D-slice imaging of bounded turbid media using a new integrodifferential equation,” Opt. Exp. 4, 231–240 (1999).
[CrossRef]

V. Ntziachristos, X. Ma, A. G. Yodh, B. Chance, “Multichannel photon counting instrument for spatially resolved near-infrared spectroscopy,” Rev. Sci. Instrum. 70, 193–201 (1999).
[CrossRef]

V. Ntziachristos, B. Chance, A. G. Yodh, “Differential diffuse optical tomography,” Opt. Exp. 5, 230–242 (1999).
[CrossRef]

X. De Li, T. Durduran, A. G. Yodh, B. Chance, D. N. Pattanayak, “Diffraction tomography for biochemical imaging with diffuse-photon density waves,” Opt. Lett. 22, 573–575 (1997).
[CrossRef]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef] [PubMed]

Zhou, G. X.

Zubkov, L.

T. Durduran, J. P. Culver, M. J. Holboke, X. D. Li, L. Zubkov, B. Chance, D. N. Pattanayak, A. G. Yodh, “Algorithms for 3D localization and imaging using near-field diffraction tomography with diffuse light,” Opt. Exp. 4, 247–262 (1999).
[CrossRef]

Appl. Opt. (10)

M. S. Patterson, B. Chance, B. C. Wilson, “Time-resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
[CrossRef] [PubMed]

S. Fantini, S. A. Walker, M. A. Franceschini, M. Kaschke, P. M. Schlag, K. T. Moesta, “Assessment of the size, position, and optical properties of breast tumors in vivo by noninvasive optical methods,” Appl. Opt. 37, 1982–1989 (1998).
[CrossRef]

A. H. Hielscher, H. Liu, B. Chance, “Time-resolved photon emission from layered turbid media,” Appl. Opt. 35, 2221–2227 (1996).
[CrossRef]

G. Alexandrakis, T. J. Farrell, M. S. Patterson, “Accuracy of the diffusion approximation in determining the optical properties of a two-layer turbid medium,” Appl. Opt. 37, 7401–7409 (1998).
[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

T. H. Pham, T. Spott, L. O. Svaasand, B. J. Tromberg, “Quantifying the properties of two-layer turbid media with frequency-domain diffuse reflectance,” Appl. Opt. 39, 4733–4745 (2000).
[CrossRef]

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[CrossRef]

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[CrossRef] [PubMed]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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Phys. Med. Biol. (1)

L. O. Svaasand, T. Spott, J. B. Frishkin, T. Pham, B. J. Tromberg, M. W. Berns, “Reflectance measurements of layered media with diffuse photon-density waves: a potential tool for evaluating deep burns and subcutaneous lesions,” Phys. Med. Biol. 44, 801–813 (1999).
[CrossRef] [PubMed]

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[CrossRef]

Rev. Sci. Instrum. (2)

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[CrossRef]

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Detailed information on the components and on how to build the resin can be found at http://www.med.upenn.edu/~oisg/oisg.html .

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Figures (7)

Fig. 1
Fig. 1

Configuration for a slab where three different zones are distinguished, with different diffusive parameters: zL, 0zL, and z0.

Fig. 2
Fig. 2

Amplitude of Rslab and Tslab at K=0 for different values of the slab width L for the cases μs1=10 cm-1, ω=0 (solid curves); μs1=20 cm-1, ω=0 (solid circles); μs1=20 cm-1, ω=200 MHz (open circles). In all cases μs0=μs2=5 cm-1, μa0=μa1=μa2=0.025 cm-1, n0=n1=n2=1.333.

Fig. 3
Fig. 3

Amplitude and phase of Rslab and Tslab at K=0 for different values of the modulation frequency ω for the cases μs1=10 cm-1 [Rslab (solid curves) and Tslab (dotted curves)], μs1=20 cm-1 [Rslab (solid circles) and Tslab (open circles)]. In all cases μs0=μs2=5 cm-1, μa0=μa1=μa2=0.025 cm-1, n0=n1=n2=1.333.

Fig. 4
Fig. 4

Multiple-layered configuration of M slabs, where nin and nout are the refractive indices of the input and output media, respectively, with both being nondiffusive. In all cases, the normal to each interface is considered to point from nin into nout, i.e., along the propagation direction of the incident wave.

Fig. 5
Fig. 5

Experimental setup.

Fig. 6
Fig. 6

Fitted values of U¯2slab(K) (solid circles) and U¯1slab(K) (open triangles), compared with the data values U¯data(K) (solid curve) versus the frequency K, for the case μs0=10 cm-1, μa0=0.075 cm-1. See Eqs. (34) and (35).

Fig. 7
Fig. 7

Fitted values with use of the two-layer expression [Eq. (34)]: (a) μs0=5 cm-1, (b) μs0=10 cm-1, (c) μs0=20 cm-1, and the one-layer expression [Eq. (35)]: (d) μs0=5 cm-1, (e) μs0=10 cm-1, (f) μs0=20 cm-1. Symbols and solid curves, the expected and the retrieved values, respectively.

Tables (1)

Tables Icon

Table 1 Comparison of the Fitted Values Obtained with Eqs. (34) and (35), Two-Layered Medium and One-Layered Medium, Respectivelya

Equations (43)

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(UR, z)=-+A(K)exp[iK·R+iq(K)|z-zs|]dK,
U(K, z)=A(K)exp[iq(K)|z-zs|],
J˜n(K, z)=-D U˜(K, z)z=-iDq(K)U˜(K, z),
U0(R, z)|z=0=αJn(R, z)|z=0=-D0α U0(R, z)zz=0,
U˜(i)(K, z=0)+U˜(r)(K, z=0)=αJ˜n(K, z=0),
J˜n(K, z=0)=-D0[iq(K)U˜(i)(K, z=0)-iq(K)U˜(r)(K, z=0)],
A(r)(K)=Rnd(K)U˜(i)(K, z=0),
Jn(K, z=0)=1α Tnd(K)U˜(i)(K, z=0),
Rnd(K)=iαD0q(K)+1iαD0q(K)-1,
Tnd(K)=2iαD0q(K)iαD0q(K)-1,
U˜(t)(K, z=0)=Tnd(K)U˜(i)(K, z=0).
U0(R, z)|z=0=U(i)(R, z)|z=0+U(r)(R, z)|z=0=0,
Rblack(K)=-1,
Tblack(K)=0.
A(i)(K)=14π2 -+U(i)(R, z=zs)exp[-iK·R]dR.
A(i)(K)=S04πD0 i2πq(K),
U˜(i)(K, z)=S04πD0 i2πqi(K) exp[iqi(K)|z-zs|].
U˜(i)(K, z)=S04πD0 i2πqi(K){exp[iqi(K)|z-zs|]-exp[iqi(K)|z+zs|]}.
U˜(i)(K, z)=S04π2D0 exp[iqi(K)z]qi(K) sinh[iqi(K)zs].
Rslab=R01+T01R12 exp[2iq1L]T10+T01R12×exp[2iq1L]R10R12×exp[2iq1L]T10+,
Tslab=T01 exp[iq1L]T10+T01R12 exp[2iq1L]R10×exp[iq1l]T10+,
A=R01+T01 exp[2iq1L]R12T101-R10R12 exp[2iq1L]U˜(i),
B=T011-R10R12 exp[2iq1L]U˜(i),
C=T01 exp[iq1L]R121-R10R12 exp[2iq1L]U˜(i),
D=T01 exp[iq1L]T121-R10R12 exp[2iq1L]U˜(i).
Rslab(K)=R01+T01 exp[2iq1L]R12T101-R10R12 exp[2iq1L],
Tslab(K)=T01 exp[iq1L]T121-R10R12 exp[2iq1L].
RslabR01+T01 exp[2iq1L]R12T10,
TslabT01 exp[iq1L]T12.
Llimit3(D1/μa1)1/2.
Uj=Aj exp[iqj(zj-z)]+Bj exp[iqj(z-zj+1)],zjzzj+1,
[Mˆ]M×M·[Xˆ]M×1=[Yˆ]M×1,
[Mˆ]fin-ginE1000000a12E1b12-1-E20000q1D1E1-q1D1-q2D2q2D2E2000000a23E2b23-1-E30000q2D2E2-q2D2-q3D3q3D3E3000000-qMDMqMDMEM000000-goutEMfout,
[Xˆ]A1B1AMBM;[Yˆ]ginU˜(i)(K, z=0)-a12U˜(i)(K, z=0)00,
Ej=exp[iqjLj],
ajk=(nk/nj)2+iCjkDjqj,
bjk=(nk/nj)2-iCjkDjqj.
fin=iαinq1D1-1;gin=iαinq1D1+1,
fout=iαoutqMDM-1;gout=iαoutqMDM+1,
U˜2layer(K, zd)=S04π2D0exp[iq0(K)L0]q0(K)×sinh[iq0(K)zs]Tresinslab(K),
U˜1layer(K, zd)=S04π2D0 exp[iq0(K)(L0+L1)]q0(K)×sinh[iq0(K)zs]Tnd(K),
f[μs0(i), μa0(j=1,,5)]=KU¯theory(i, j=1,,5)(K)-U¯data(i, j=1,,5)(K)2,
Δμa0¯=15 j=15|μa0found(j)-μa0expected(j)|[cm-1],

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