Abstract

The scattering-phase theorem states that the values of scattering and reduced scattering coefficients of the bulk random media are proportional to the variance of the phase and the variance of the phase gradient, respectively, of the phase map of light passing through one thin slice of the medium. We report a new derivation of the scattering phase theorem and provide the correct form of the relation between the variance of phase gradient and the reduced scattering coefficient. We show the scattering-phase theorem is the consequence of anomalous diffraction by a thin slice of forward-peaked scattering media. A new set of scattering-phase relations with relaxed requirement on the thickness of the slice are provided. The condition for the scattering-phase theorem to be valid is discussed and illustrated with simulated data. The scattering-phase theorem is then applied to determine the scattering coefficient μs, the reduced scattering coefficient μs, and the anisotropy factor g for polystyrene sphere and Intralipid-20% suspensions with excellent accuracy from quantitative phase imaging of respective thin slices. The spatially-resolved μs, μs and g maps obtained via such a scattering-phase relationship may find general applications in the characterization of the optical property of homogeneous and heterogeneous random media.

© 2011 OSA

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    [CrossRef]
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2011

2010

2008

W. S. Rockward, A. L. Thomas, B. Zhao, and C. A. DiMarzio, “Quantitative phase measurements using optical quadrature microscopy,” Appl. Opt. 47(10), 1684–1696 (2008).
[CrossRef] [PubMed]

H. Ding, Z. Wang, F. Nguyen, S. A. Boppart, and G. Popescu, “Fourier Transform Light Scattering of Inhomogeneous and Dynamic Structures,” Phys. Rev. Lett. 101(23), 238102 (2008).
[CrossRef] [PubMed]

M. Xu, T. T. Wu, and J. Y. Qu, “Unified Mie and fractal scattering by cells and experimental study on application in optical characterization of cellular and subcellular structures,” J. Biomed. Opt. 13, 038802 (2008).

2007

2006

2005

M. Xu and R. R. Alfano, “Fractal mechanisms of light scattering in biological tissue and cells,” Opt. Lett. 30, 3051–3053 (2005).
[CrossRef] [PubMed]

S. Menon, Q. Su, and R. Grobe, “Determination of g and ? using multiply scattered light in turbid media,” Phys. Rev. Lett. 94, 153904 (2005).
[CrossRef] [PubMed]

2004

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy.” J. Microsc. 214(1), 7–12 (2004).
[CrossRef] [PubMed]

P. Yang, Z. Zhang, B. Baum, H.-L. Huang, and Y. Hu, “A new look at anomalous diffraction theory (ADT): Algorithm in cumulative projected-area distribution domain and modified ADT,” J. Quant. Spectrosc. Radiat. Transf. 89, 421–442 (2004).
[CrossRef]

2003

1998

1996

1991

1990

W. F. Cheong, S. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

1987

S. A. Ackerman and G. L. Stephens, “The absorption of solar radiation by cloud droplets: an application of anomalous diffraction theory,” J. Atmos. Sci. 44(12), 1574–1588 (1987).
[CrossRef]

1980

Ackerman, S. A.

S. A. Ackerman and G. L. Stephens, “The absorption of solar radiation by cloud droplets: an application of anomalous diffraction theory,” J. Atmos. Sci. 44(12), 1574–1588 (1987).
[CrossRef]

Alfano, R. R.

Alrubaiee, M.

M. Xu, M. Alrubaiee, and R. R. Alfano, “Fractal mechanism of light scattering for tissue optical biopsy,” in Optical Biopsy VI, R. R. Alfano and A. Katz, eds., vol. 6091 of Proceedings of SPIE, p. 60910E (2006).

Arnison, M. R.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy.” J. Microsc. 214(1), 7–12 (2004).
[CrossRef] [PubMed]

Arzumanov, G.

M. Iftikhar, B. DeAngelo, G. Arzumanov, P. Shanley, Z. Xu, and M. Xu, “Characterizing scattering property of random media from phase map of a thin slice: the scattering-phase theorem and the intensity propagation equation approach,” in Optical Tomography and Spectroscopy of Tissue IX, B. J. Tromberg, A. G. Yodh, M. Tamura, E. M. Sevick-Muraca, and R. R. Alfano, eds., vol. 7896 of Proceedings of SPIE, p. 78961O (SPIE, Bellingham, WA, 2011).

Backman, V.

Barbastathis, G.

Barty, A.

Baum, B.

P. Yang, Z. Zhang, B. Baum, H.-L. Huang, and Y. Hu, “A new look at anomalous diffraction theory (ADT): Algorithm in cumulative projected-area distribution domain and modified ADT,” J. Quant. Spectrosc. Radiat. Transf. 89, 421–442 (2004).
[CrossRef]

Boppart, S. A.

H. Ding, Z. Wang, F. Nguyen, S. A. Boppart, and G. Popescu, “Fourier Transform Light Scattering of Inhomogeneous and Dynamic Structures,” Phys. Rev. Lett. 101(23), 238102 (2008).
[CrossRef] [PubMed]

Borghese, F.

Cheong, W. F.

W. F. Cheong, S. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Chýlek, P.

Cogswell, C. J.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy.” J. Microsc. 214(1), 7–12 (2004).
[CrossRef] [PubMed]

Dasari, R. R.

DeAngelo, B.

M. Iftikhar, B. DeAngelo, G. Arzumanov, P. Shanley, Z. Xu, and M. Xu, “Characterizing scattering property of random media from phase map of a thin slice: the scattering-phase theorem and the intensity propagation equation approach,” in Optical Tomography and Spectroscopy of Tissue IX, B. J. Tromberg, A. G. Yodh, M. Tamura, E. M. Sevick-Muraca, and R. R. Alfano, eds., vol. 7896 of Proceedings of SPIE, p. 78961O (SPIE, Bellingham, WA, 2011).

Denti, P.

DiMarzio, C. A.

Ding, H.

Z. Wang, H. Ding, and G. Popescu, “Scattering-phase theorem.” Opt. Lett. 36(7), 1215–1217 (2011).
[CrossRef] [PubMed]

H. Ding, Z. Wang, F. Nguyen, S. A. Boppart, and G. Popescu, “Fourier Transform Light Scattering of Inhomogeneous and Dynamic Structures,” Phys. Rev. Lett. 101(23), 238102 (2008).
[CrossRef] [PubMed]

Z. Wang, H. Ding, K. Tangella, and G. Popescu, “A scattering-phase theorem,” in Biomedical Optics (BIOMED) Topical Meeting and Tabletop Exhibit, p. BTuD111p (OSA, 2010).

Feld, M. S.

Giusto, A.

Gneiting, T.

P. Guttorp and T. Gneiting, “On the Whittle-Matrn correlation family,” Tech. Rep. NRCSE-TRS No. 080, NRCSE, University of Washington (2005).

Grobe, R.

S. Menon, Q. Su, and R. Grobe, “Determination of g and ? using multiply scattered light in turbid media,” Phys. Rev. Lett. 94, 153904 (2005).
[CrossRef] [PubMed]

Guttorp, P.

P. Guttorp and T. Gneiting, “On the Whittle-Matrn correlation family,” Tech. Rep. NRCSE-TRS No. 080, NRCSE, University of Washington (2005).

Hu, Y.

P. Yang, Z. Zhang, B. Baum, H.-L. Huang, and Y. Hu, “A new look at anomalous diffraction theory (ADT): Algorithm in cumulative projected-area distribution domain and modified ADT,” J. Quant. Spectrosc. Radiat. Transf. 89, 421–442 (2004).
[CrossRef]

Huang, H.-L.

P. Yang, Z. Zhang, B. Baum, H.-L. Huang, and Y. Hu, “A new look at anomalous diffraction theory (ADT): Algorithm in cumulative projected-area distribution domain and modified ADT,” J. Quant. Spectrosc. Radiat. Transf. 89, 421–442 (2004).
[CrossRef]

Iat, M. A.

Iftikhar, M.

M. Iftikhar, B. DeAngelo, G. Arzumanov, P. Shanley, Z. Xu, and M. Xu, “Characterizing scattering property of random media from phase map of a thin slice: the scattering-phase theorem and the intensity propagation equation approach,” in Optical Tomography and Spectroscopy of Tissue IX, B. J. Tromberg, A. G. Yodh, M. Tamura, E. M. Sevick-Muraca, and R. R. Alfano, eds., vol. 7896 of Proceedings of SPIE, p. 78961O (SPIE, Bellingham, WA, 2011).

Ikeda, T.

Katz, A.

M. Xu and A. Katz, Light Scattering Reviews, vol. III, chap. Statistical Interpretation of Light Anomalous Diffraction by Small Particles and its Applications in Bio-agent Detection and Monitoring, pp. 27–68 (Springer, 2008).
[CrossRef]

Klett, J. D.

Kou, S. S.

Kumar, G.

Larkin, K. G.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy.” J. Microsc. 214(1), 7–12 (2004).
[CrossRef] [PubMed]

Lax, M.

Ma, S. H.

S. H. Ma, Statistical Mechanics (World Scientific, 1985).

Menon, S.

S. Menon, Q. Su, and R. Grobe, “Determination of g and ? using multiply scattered light in turbid media,” Phys. Rev. Lett. 94, 153904 (2005).
[CrossRef] [PubMed]

Moes, C. J. M.

Nguyen, F.

H. Ding, Z. Wang, F. Nguyen, S. A. Boppart, and G. Popescu, “Fourier Transform Light Scattering of Inhomogeneous and Dynamic Structures,” Phys. Rev. Lett. 101(23), 238102 (2008).
[CrossRef] [PubMed]

Nugent, K. A.

Paganin, D.

Popescu, G.

Z. Wang, H. Ding, and G. Popescu, “Scattering-phase theorem.” Opt. Lett. 36(7), 1215–1217 (2011).
[CrossRef] [PubMed]

H. Ding, Z. Wang, F. Nguyen, S. A. Boppart, and G. Popescu, “Fourier Transform Light Scattering of Inhomogeneous and Dynamic Structures,” Phys. Rev. Lett. 101(23), 238102 (2008).
[CrossRef] [PubMed]

G. Popescu, T. Ikeda, R. R. Dasari, and M. S. Feld, “Diffraction phase microscopy for quantifying cell structure and dynamics,” Opt. Lett. 31(6), 775–777 (2006).
[CrossRef] [PubMed]

Z. Wang, H. Ding, K. Tangella, and G. Popescu, “A scattering-phase theorem,” in Biomedical Optics (BIOMED) Topical Meeting and Tabletop Exhibit, p. BTuD111p (OSA, 2010).

Prahl, S.

W. F. Cheong, S. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Prahl, S. A.

Qu, J. Y.

M. Xu, T. T. Wu, and J. Y. Qu, “Unified Mie and fractal scattering by cells and experimental study on application in optical characterization of cellular and subcellular structures,” J. Biomed. Opt. 13, 038802 (2008).

T. T. Wu, J. Y. Qu, and M. Xu, “Unified Mie and fractal scattering by biological cells and subcellular structures,” Opt. Lett. 32, 2324–2326 (2007).
[CrossRef] [PubMed]

Radosevich, A.

Roberts, A.

Rockward, W. S.

Rogers, J. D.

Saija, R.

Schlather, M.

M. Schlather, “An introduction to positive-definite functions and to unconditional simulation of random fields,” Tech. Rep. ST-99-10, Lancaster University (1999).

Schmitt, J. M.

Shanley, P.

M. Iftikhar, B. DeAngelo, G. Arzumanov, P. Shanley, Z. Xu, and M. Xu, “Characterizing scattering property of random media from phase map of a thin slice: the scattering-phase theorem and the intensity propagation equation approach,” in Optical Tomography and Spectroscopy of Tissue IX, B. J. Tromberg, A. G. Yodh, M. Tamura, E. M. Sevick-Muraca, and R. R. Alfano, eds., vol. 7896 of Proceedings of SPIE, p. 78961O (SPIE, Bellingham, WA, 2011).

Sheppard, C. J. R.

S. S. Kou, L. Waller, G. Barbastathis, and C. J. R. Sheppard, “Transport-of-intensity approach to differential interference contrast (TI-DIC) microscopy for quantitative phase imaging,” Opt. Lett. 35(3), 447–449 (2010).
[CrossRef] [PubMed]

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy.” J. Microsc. 214(1), 7–12 (2004).
[CrossRef] [PubMed]

Sindoni, O. I.

Smith, N. I.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy.” J. Microsc. 214(1), 7–12 (2004).
[CrossRef] [PubMed]

Stephens, G. L.

S. A. Ackerman and G. L. Stephens, “The absorption of solar radiation by cloud droplets: an application of anomalous diffraction theory,” J. Atmos. Sci. 44(12), 1574–1588 (1987).
[CrossRef]

Su, Q.

S. Menon, Q. Su, and R. Grobe, “Determination of g and ? using multiply scattered light in turbid media,” Phys. Rev. Lett. 94, 153904 (2005).
[CrossRef] [PubMed]

Taflove, A.

Tangella, K.

Z. Wang, H. Ding, K. Tangella, and G. Popescu, “A scattering-phase theorem,” in Biomedical Optics (BIOMED) Topical Meeting and Tabletop Exhibit, p. BTuD111p (OSA, 2010).

Thomas, A. L.

Turzhitsky, V.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

van GemertJ, M. J. C.

van Marle, J.

van Staveren, H. J.

Waller, L.

Wang, Z.

Z. Wang, H. Ding, and G. Popescu, “Scattering-phase theorem.” Opt. Lett. 36(7), 1215–1217 (2011).
[CrossRef] [PubMed]

H. Ding, Z. Wang, F. Nguyen, S. A. Boppart, and G. Popescu, “Fourier Transform Light Scattering of Inhomogeneous and Dynamic Structures,” Phys. Rev. Lett. 101(23), 238102 (2008).
[CrossRef] [PubMed]

Z. Wang, H. Ding, K. Tangella, and G. Popescu, “A scattering-phase theorem,” in Biomedical Optics (BIOMED) Topical Meeting and Tabletop Exhibit, p. BTuD111p (OSA, 2010).

Welch, A. J.

W. F. Cheong, S. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Wiscombe, W. J.

Wu, T. T.

M. Xu, T. T. Wu, and J. Y. Qu, “Unified Mie and fractal scattering by cells and experimental study on application in optical characterization of cellular and subcellular structures,” J. Biomed. Opt. 13, 038802 (2008).

T. T. Wu, J. Y. Qu, and M. Xu, “Unified Mie and fractal scattering by biological cells and subcellular structures,” Opt. Lett. 32, 2324–2326 (2007).
[CrossRef] [PubMed]

Xu, M.

M. Xu, T. T. Wu, and J. Y. Qu, “Unified Mie and fractal scattering by cells and experimental study on application in optical characterization of cellular and subcellular structures,” J. Biomed. Opt. 13, 038802 (2008).

T. T. Wu, J. Y. Qu, and M. Xu, “Unified Mie and fractal scattering by biological cells and subcellular structures,” Opt. Lett. 32, 2324–2326 (2007).
[CrossRef] [PubMed]

M. Xu and R. R. Alfano, “Fractal mechanisms of light scattering in biological tissue and cells,” Opt. Lett. 30, 3051–3053 (2005).
[CrossRef] [PubMed]

M. Xu, M. Lax, and R. R. Alfano, “Light anomalous diffraction using geometrical path statistics of rays and Gaussian ray approximation,” Opt. Lett. 28, 179–181 (2003).
[CrossRef] [PubMed]

M. Iftikhar, B. DeAngelo, G. Arzumanov, P. Shanley, Z. Xu, and M. Xu, “Characterizing scattering property of random media from phase map of a thin slice: the scattering-phase theorem and the intensity propagation equation approach,” in Optical Tomography and Spectroscopy of Tissue IX, B. J. Tromberg, A. G. Yodh, M. Tamura, E. M. Sevick-Muraca, and R. R. Alfano, eds., vol. 7896 of Proceedings of SPIE, p. 78961O (SPIE, Bellingham, WA, 2011).

M. Xu, M. Alrubaiee, and R. R. Alfano, “Fractal mechanism of light scattering for tissue optical biopsy,” in Optical Biopsy VI, R. R. Alfano and A. Katz, eds., vol. 6091 of Proceedings of SPIE, p. 60910E (2006).

M. Xu and A. Katz, Light Scattering Reviews, vol. III, chap. Statistical Interpretation of Light Anomalous Diffraction by Small Particles and its Applications in Bio-agent Detection and Monitoring, pp. 27–68 (Springer, 2008).
[CrossRef]

Xu, Z.

M. Iftikhar, B. DeAngelo, G. Arzumanov, P. Shanley, Z. Xu, and M. Xu, “Characterizing scattering property of random media from phase map of a thin slice: the scattering-phase theorem and the intensity propagation equation approach,” in Optical Tomography and Spectroscopy of Tissue IX, B. J. Tromberg, A. G. Yodh, M. Tamura, E. M. Sevick-Muraca, and R. R. Alfano, eds., vol. 7896 of Proceedings of SPIE, p. 78961O (SPIE, Bellingham, WA, 2011).

Yang, P.

P. Yang, Z. Zhang, B. Baum, H.-L. Huang, and Y. Hu, “A new look at anomalous diffraction theory (ADT): Algorithm in cumulative projected-area distribution domain and modified ADT,” J. Quant. Spectrosc. Radiat. Transf. 89, 421–442 (2004).
[CrossRef]

Zhang, Z.

P. Yang, Z. Zhang, B. Baum, H.-L. Huang, and Y. Hu, “A new look at anomalous diffraction theory (ADT): Algorithm in cumulative projected-area distribution domain and modified ADT,” J. Quant. Spectrosc. Radiat. Transf. 89, 421–442 (2004).
[CrossRef]

Zhao, B.

Appl. Opt.

Biomed. Opt. Express

IEEE J. Quantum Electron.

W. F. Cheong, S. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

J. Atmos. Sci.

S. A. Ackerman and G. L. Stephens, “The absorption of solar radiation by cloud droplets: an application of anomalous diffraction theory,” J. Atmos. Sci. 44(12), 1574–1588 (1987).
[CrossRef]

J. Biomed. Opt.

M. Xu, T. T. Wu, and J. Y. Qu, “Unified Mie and fractal scattering by cells and experimental study on application in optical characterization of cellular and subcellular structures,” J. Biomed. Opt. 13, 038802 (2008).

J. Microsc.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy.” J. Microsc. 214(1), 7–12 (2004).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

J. Quant. Spectrosc. Radiat. Transf.

P. Yang, Z. Zhang, B. Baum, H.-L. Huang, and Y. Hu, “A new look at anomalous diffraction theory (ADT): Algorithm in cumulative projected-area distribution domain and modified ADT,” J. Quant. Spectrosc. Radiat. Transf. 89, 421–442 (2004).
[CrossRef]

Opt. Lett.

Phys. Rev. Lett.

S. Menon, Q. Su, and R. Grobe, “Determination of g and ? using multiply scattered light in turbid media,” Phys. Rev. Lett. 94, 153904 (2005).
[CrossRef] [PubMed]

H. Ding, Z. Wang, F. Nguyen, S. A. Boppart, and G. Popescu, “Fourier Transform Light Scattering of Inhomogeneous and Dynamic Structures,” Phys. Rev. Lett. 101(23), 238102 (2008).
[CrossRef] [PubMed]

Other

S. H. Ma, Statistical Mechanics (World Scientific, 1985).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

The factor i appears in Eq. (2) and did not appear in anomalous diffraction by optically soft particles described by Hulst in Ref 12. This difference originates from the fact that the scattering wave into direction ? is proportional to ?iS(?) in the Hulst convention and S(?) in the contemporary convention adopted here.

M. Iftikhar, B. DeAngelo, G. Arzumanov, P. Shanley, Z. Xu, and M. Xu, “Characterizing scattering property of random media from phase map of a thin slice: the scattering-phase theorem and the intensity propagation equation approach,” in Optical Tomography and Spectroscopy of Tissue IX, B. J. Tromberg, A. G. Yodh, M. Tamura, E. M. Sevick-Muraca, and R. R. Alfano, eds., vol. 7896 of Proceedings of SPIE, p. 78961O (SPIE, Bellingham, WA, 2011).

Z. Wang, H. Ding, K. Tangella, and G. Popescu, “A scattering-phase theorem,” in Biomedical Optics (BIOMED) Topical Meeting and Tabletop Exhibit, p. BTuD111p (OSA, 2010).

M. Xu and A. Katz, Light Scattering Reviews, vol. III, chap. Statistical Interpretation of Light Anomalous Diffraction by Small Particles and its Applications in Bio-agent Detection and Monitoring, pp. 27–68 (Springer, 2008).
[CrossRef]

P. Guttorp and T. Gneiting, “On the Whittle-Matrn correlation family,” Tech. Rep. NRCSE-TRS No. 080, NRCSE, University of Washington (2005).

M. Schlather, “An introduction to positive-definite functions and to unconditional simulation of random fields,” Tech. Rep. ST-99-10, Lancaster University (1999).

M. Xu, M. Alrubaiee, and R. R. Alfano, “Fractal mechanism of light scattering for tissue optical biopsy,” in Optical Biopsy VI, R. R. Alfano and A. Katz, eds., vol. 6091 of Proceedings of SPIE, p. 60910E (2006).

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Figures (2)

Fig. 2
Fig. 2

The optical path length map ΔOPL for a monolayer of polystyrene sphere suspension (size: 8.31μm) in water (left) and a thin film (thickness: 4μm) of Intralipid-20% suspension (right).

Fig. 1
Fig. 1

The normalized phase map ( Δ OPL / δ m 2 l), the ratio of 2〈1 – cosΔϕ〉 over μsL, and the ratio of (2k2)–1 〈|∇ϕ|2〉 over μsL are displayed, from left- to right-hand direction, for a thin slice of random medium of varying thickness L with the refractive index fluctuation following the Whittle-Matern correlation function of ν = 1.0 (top row), ν = 0.5 (middle row) and ν = 0.1 (bottom row). The normalized phase map is shown for L = 20l.

Equations (17)

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μ s L = ( Δ ϕ ) 2
S ( θ ) = i k 2 2 π 1 + cos θ 2 ( 1 e i Δ ϕ ( ρ ) ) exp ( iks ρ ) d ρ
C sca = 2 ( 1 cos Δ ϕ ) d ρ .
C sca = 1 8 π 2 d s d ρ ( 1 e i Δ ϕ ( ρ ) ) d d ρ exp ( ik s ρ ) × d ρ ( 1 e i Δ ϕ ( ρ ) ) d d ρ exp ( ik s ρ ) .
C sca = 1 2 k 2 d ρ d ρ [ d d ρ ( 1 e i Δ ϕ ( ρ ) ) ] [ d d ρ ( 1 e i Δ ϕ ( ρ ) ) ] δ ( ρ ρ ) = 1 2 k 2 | d d ρ ( 1 e i Δ ϕ ( ρ ) ) | 2 d ρ ,
C sca = 1 2 k 2 | ϕ | 2 d ρ .
μ s L = 2 A ( 1 cos Δ ϕ ) d ρ = 2 1 cos Δ ϕ
μ s L = 1 2 k 2 A | ϕ | 2 d ρ = 1 2 k 2 | ϕ | 2
g = 1 | ϕ | 2 4 k 2 ( 1 cos Δ ϕ ) 1 | ϕ | 2 2 k 2 ( Δ ϕ ) 2 .
R n ( r ) = ( δ m ) 2 γ ( r l )
γ ( r l ) = 2 1 ν | Γ ( ν ) | 1 ( r l ) ν K ν ( r l )
R ^ n ( q ) = ( δ m ) 2 Γ ( ν + 3 / 2 ) π 3 / 2 | Γ ( ν ) | l 3 ( 1 + q 2 l 2 ) ν 3 / 2
μ s = 2 π 1 / 2 k 2 l ( δ m ) 2 Γ ( ν + 1 / 2 ) | Γ ( ν ) | [ 1 ( 1 + 4 X 2 ) ν 1 / 2 ] ,
μ s = π 1 / 2 l 1 ( δ m ) 2 Γ ( ν + 1 / 2 ) | Γ ( ν ) | 1 ν 1 / 2 × [ 1 ( 1 + 4 X 2 ) ν 1 / 2 [ 1 + 4 X 2 ( ν + 1 / 2 ) ] ] ,
Δ ϕ ( ρ ) Δ ϕ ( ρ ) = μ s L 1 2 ( k Δ ρ ) 2 μ s L
P ( q ) = | U ˜ ( q ) | 2 | U ˜ ( q ) | 2 dq
U ˜ ( q ) = 1 ( 2 π ) 2 [ e i Δ ϕ ( ρ ) 1 ] exp ( i ρ q ) d ρ .

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