## Scattering-phase theorem: anomalous diffraction by forward-peaked scattering media |

Optics Express, Vol. 19, Issue 22, pp. 21643-21651 (2011)

http://dx.doi.org/10.1364/OE.19.021643

Acrobat PDF (1186 KB)

### 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 *μ _{s}*, the reduced scattering coefficient

*μ*′

*, and the anisotropy factor*

_{s}*g*for polystyrene sphere and Intralipid-20% suspensions with excellent accuracy from quantitative phase imaging of respective thin slices. The spatially-resolved

*μ*,

_{s}*μ*′

*and*

_{s}*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

## 1. Introduction

*μ*′

*The transport mean free path, given by the inverse of*

_{s}*μ*′

*, can be significantly larger than the distance that light travels between consecutive scattering events,*

_{s}1. 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]

*ϕ*of the transmitted light wave can be measured using quantitative phase imaging. The two extreme cases of light propagation in random media–diffusion of multiply scattered light and transmission of minimally scattered light–has been recently suggested inherently connected first by Wang et al. [2, 3

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

4. 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).

*μ*and

_{s}*μ*′

*of the bulk media are found to be 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. This is so called “scattering-phase theorem.”*

_{s}*μ*′

*. The anisotropy factor,*

_{s}*g*≡ 1 –

*μ*′

*/*

_{s}*μ*, an important parameter linked to the morphology of the scatterers in the medium, can then be derived directly from the phase map. More importantly, we show the scattering-phase theorem is the consequence of anomalous diffraction by a thin slice of forward-peaked scattering media. A set of

_{s}*μ*–

_{s}*ϕ*,

*μ*′

*–*

_{s}*ϕ*, and

*g*–

*ϕ*relations are provided, for the first time, with relaxed requirement on the thickness of the slice. 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, the reduced scattering coefficient and the anisotropy factor for polystyrene sphere and Intralipid-20% suspensions with excellent accuracy from their quantitative phase maps measured by differential interference contrast microscopy. The paper ends with a discussion of the significance and applications of this scattering-phase relationship.

## 2. Theory

*L*illuminated by a plane wave of unit intensity. The spatially resolved phase map

*ϕ*(

*ρ*) for wave transmission is expressed as

*k*≡ 2

*πn*

_{0}/

*λ*is the wave number,

*n*

_{0}is the background refractive index,

*λ*is the wavelength of light in vacuum, and

*m*is the relative refractive index at position (

*ρ*,

*z*) with

*ρ*and

*z*the lateral and axial coordinates, respectively. The fluctuation in relative refractive index

*δm*≡

*m*– 1 satisfies 〈

*δm*〉 = 0 where 〈〉 means the spatial average. The phase map

*ϕ*(

*ρ*) can be readily measured with quantitative phase imaging approaches [5

5. A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy.” Opt. Lett. **23**(11), 817–819 (1998). [CrossRef]

9. 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]

*μ*of the bulk medium and the variance of the phase has been obtained based on the decomposition of the transmitted statistically homogeneous wave field

_{s}*U*into its spatial average and a spatially varying component

*U*(

*ρ*) =

*U*

_{0}(

*ρ*) +

*U*

_{1}(

*ρ*) and the fact that

*U*

_{0}= 〈

*U*〉 corresponds to the unscattered wave and

*U*

_{1}is the scattered component [2, 10

10. 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]

*U*

_{0}|

^{2}= |〈

*e*

^{iϕ(ρ)}〉|

^{2}= exp(−

*μ*) by the Beer’s law. Hence

_{s}L*μ*= −2ln|〈

_{s}L*e*

^{iΔϕ(ρ)}〉| where

*ϕ*| ≪ 1 as implied by

*μ*and the variance of the phase, and the reduced scattering coefficient

_{s}*μ*′

*and the variance of the phase gradient are the consequence of anomalous diffraction by a thin slice of forward-peaked scattering media and the requirement of*

_{s}*θ*due to the thin slice is given by where

*ks*is the propagation direction of the scattered light and

*s*is a unit direction vector using the Huygens’ principle [12, 13

13. 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 *θ* is proportional to −*iS*(*θ*) in the Hulst convention and *S*(*θ*) in the contemporary convention adopted here.

*z*=

*L*plane to

*e*

^{iΔϕ(ρ)}from 1 and hence the

*scattered*wave is

*e*

^{iΔϕ(ρ)}– 1 whereas

*e*

^{iΔϕ(ρ)}is the total wave on that plane [12]. We could replace cos

*θ*in Eq. (2) by 1 as scattering is forward-peaked. The scattering cross section

*C*

_{sca}= 4

*πk*

^{−2}ℑ

*S*(0) by the optical extinction theorem is then found to be The reduced scattering cross section

*C*′

_{sca}=

*k*

^{−2}∫(1 – cos

*θ*)|

*S*(

*θ*)|

^{2}

*d*Ω can be simplified by first writing

*s*

_{⊥}is the projection of

*s*on the lateral plane and rewriting

*C*′

_{sca}as By performing partial integration on

*d*/

*dρ*and

*d*/

*dρ*′ in Eq. (4) and then integrating over

*s*

_{⊥}, we have which reduces to As the scattering and the reduced scattering cross sections are given by

*μ*and

_{s}AL*μ*′

*, respectively, by definition for the thin slice of area*

_{s}AL*A*, we obtain and from Eqs. (3) and (6). In addition, the anisotropy factor

*g*≡ 1 –

*μ*′

*/*

_{s}*μ*, representing the mean cosine of the scattering angle, is given by

_{s}## 3. Simulations and experiments

*R*(

_{n}**r**) = 〈

*δm*(

**r**′)

*δm*(

**r**′ +

**r**)〉 is assumed to be the Whittle-Matern correlation function [19] given by: with where

*K*(·) is the modified Bessel function of the second kind. The Whittle-Matern correlation function has been used extensively to model turbulence and refractive index fluctuation in biological tissue [20

_{ν}20. J. M. Schmitt and G. Kumar, “Turbulent nature of refractive-index variations in biological tissue,” Opt. Lett. **21**, 1310–1312 (1996). [CrossRef] [PubMed]

21. V. Turzhitsky, A. Radosevich, J. D. Rogers, A. Taflove, and V. Backman, “A predictive model of backscattering at subdiffusion length scales,” Biomed. Opt. Express **1**(3), 1034–1046 (2010), URL http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-1-3-1034.

*δm*)〉

^{2}= 0.01

^{2},

*l*∼ 0.5

*μm*, and

*n*

_{0}= 1.367 for biological tissue [22, 23

23. 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]

*ν*> −3/2. Light scattering by the random medium is fully described by the power spectrum of the fluctuation of the refractive index. Following [22,24

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

*X*≡

*kl*is the size parameter.

*δm*)

^{2}〉 = 0.01

^{2}, the correlation length

*l*= 0.5

*μm*, the background refractive index of the sample

*n*

_{0}= 1.367, and the wavelength of the incident beam

*λ*= 0.5

*μm*in the simulation. The random field inside a box of size 10

*l*× 10

*l*×

*L*with varying thickness

*L*=

*l*, 5

*l*, 20

*l*, and 100

*l*was simulated using RandomFields [25] with a specified spacial resolution. The phase map was generated by line integration. The gradient of the phase was computed from the phase map using the finite difference between neighboring phases. Total 15 simulations were performed for each set of parameters with their mean and standard deviation being reported hereafter.

*ϕ*〉 over

*μ*, and the ratio of (2

_{s}L*k*

^{2})

^{−1}〈|∇

*ϕ*|

^{2}〉 over

*μ*′

*for various*

_{s}L*ν*. The normalized phase map is shown for thin slices of thickness

*L*= 20

*l*. The scattering coefficient and the anisotropy factor for the bulk random medium are,

*μ*= 0.023 and

_{s}l*g*= 0.988 in the case of

*ν*= 1.0,

*μ*= 0.015 and

_{s}l*g*= 0.968 in the case of

*ν*= 0.5, and

*μ*= 0.0040 and

_{s}l*g*= 0.915 in the case of

*ν*= 0.1, respectively. The thickness of the samples covers the range starting from

*μ*≪ 1 to

_{s}L*μ*> 1. The two ratios 2〈1 – cosΔ

_{s}L*ϕ*〉/

*μ*and (2

_{s}L*k*

^{2})

^{−1}〈|∇

*ϕ*|

^{2}〉/

*μ*′

*are expected to be unity according to Eqs. (7) and (8). Figure 3 shows the former ratio approaches unity when the thickness of the medium is at least 5*

_{s}L*l*. The value of

*μ*can be computed from the phase map at all levels of resolution. On the other hand, the resolution matters for probing

_{s}*μ*′

*. The latter ratio approaches unity and the best estimation for*

_{s}*μ*′

*is obtained only when the resolution is 0.1*

_{s}*l*− 0.2

*l*and the thickness

*L*≥ 5

*l*. Insufficient resolution results in an underestimation of

*μ*′

*.*

_{s}*μm*using Canon 5D Mark II. The quantitative phase map for a monolayer of polystyrene sphere suspension (size: 8.31

*μm*) in water and a thin film (thickness: 4

*μm*) of Intralipid-20% suspension on a glass microscope slide were computed from in-focus and out-of-focus (

*δz*= 1

*μm*) DIC images under Köhler illumination using the transport-of-intensity approach [9

9. 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]

*OPL*for the two samples. The scattering property for each individual spheres can be analyzed by applying the scattering-phase theorem to the region in the phase map being occupied by the sphere. For example, the region highlighted by white dash lines for the central sphere yields

*μ*= 0.234

_{s}*μm*

^{−1},

*μ*′

*= 0.0202*

_{s}*μm*

^{−1}and

*g*= 0.91 with an area 61.0

*μm*

^{2}. The scattering and reduced scattering cross sections are 118

*μm*

^{2}and 10.2

*μm*

^{2}. The mean scattering and reduced scattering cross sections for all the spheres contained in the displayed section are 116

*μm*

^{2}and 9.8

*μm*

^{2}, respectively. These values are in excellent agreement with the theoretical prediction for a polystyrene sphere of the specified size (

*C*

_{sca}= 125

*μm*

^{2},

*C*′

_{sca}= 9.5

*μm*

^{2}and

*g*= 0.92) computed with a Mie code [26

26. W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. **19**, 1505–1509 (1980). [CrossRef] [PubMed]

*μm*

^{−1}and 0.001

*μm*

^{−1}, respectively, from the whole section displayed in Fig. 2. The former agrees with the known

*μ*value (0.139

_{s}*μm*

^{−1}) whereas the latter dramatically underestimates

*μ*′

*(0.031*

_{s}*μm*

^{−1}) at 550

*nm*[27

27. H. J. van Staveren, C. J. M. Moes, J. van Marle, S. A. Prahl, and M. J. C. van GemertJ, “Light scattering in Intralipid-10% in the wavelength range of 400–1100nm,” Appl. Opt. **30**(31), 4507–4514 (1991). [CrossRef] [PubMed]

28. A. Giusto, R. Saija, M. A. Iat, P. Denti, F. Borghese, and O. I. Sindoni, “Optical properties of high-density dispersions of particles: application to intralipid solutions,” Appl. Opt. **42**(21), 4375–4380 (2003). [CrossRef] [PubMed]

*μ*′

*directly (see Fig. 1). The quality of*

_{s}*μ*′

*estimation, however, can be significantly improved by properly taking into account light diffraction in the microscope and sharpening the phase map accordingly. This procedure yields the new value of*

_{s}*μ*′

*to be 0.022*

_{s}*μm*

^{−1}, agreeing reasonably well with the real value. The detail will be published elsewhere.

## 4. Discussion

*μ*–

_{s}*ϕ*and

*μ*′

*–*

_{s}*ϕ*relations can be justified intuitively as the following. Light scattering (

*μ*) depends on the fluctuation of the refractive index which emerges as the variance in the phase map for light transmission through a thin slice. Light reduced scattering (

_{s}*μ*′

*) reflects the deviation of the equal-phase wave front away from the forward direction which is described by the local tilt (gradient) in the phase for light transmission through a thin slice. Assuming the thin slice of sample of thickness*

_{s}*L*is uniformly divided into

*N*=

*L/l*layers with

*l*the correlation length of the random medium, Δ

*ϕ*(and ∇

*ϕ*) is the summation of

*N*independent random numbers from the

*N*layers. Hence the spatial average 〈(Δ

*ϕ*)

^{2}〉 (and 〈|∇

*ϕ*|

^{2}〉) scales with

*N*rather than

*N*

^{2}. These considerations lead to

*μ*∝ (Δ

_{s}L*ϕ*)

^{2}and

*μ*′

*∝ 〈|∇*

_{s}L*ϕ*|

^{2}〉. In cases such as a monolayer of scatterers of size much larger than the wavelength, the condition that |Δ

*ϕ*| ≪ 1 is not satisfied, the more general

*μ*–

_{s}*ϕ*relation (7) should be used whereas the

*μ*′

*–*

_{s}*ϕ*relation remains the same provided the rays do not deviate from the forward direction (the scatterers are optically soft).

*l*(with a sufficient large

*N*) to obtain the values of

*μ*and

_{s}*μ*′

*correctly from the phase map as observed in the simulation. To properly compute the local tilt in the phase to obtain*

_{s}*μ*′

*with finite difference, the separation between the two points must be smaller than the size of the scattering structure. The separation at the order of 0.1*

_{s}*l*– 0.2

*l*may be optimal as suggested by the simulation.

*ϕ*| ≪ 1 has also been obtained previously by us using another approach [4

4. 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).

*ϕ*(

*ρ*)Δ

*ϕ*(

*ρ*′)〉 between two points

*ρ*and

*ρ*′ on the phase map for light transmission through a thin slice of a weakly scattering random medium. The scattering-phase theorem in this limit is equivalent to where Δ

*ρ*≡ |

*ρ*–

*ρ*′| is the distance between the two points [4

4. 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).

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

*g*–

*ϕ*relation. The difference originates from the scattered wave was assumed to be

*e*

^{iΔϕ(ρ)}in our notation in Ref [2, 3

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

*z*=

*L*plane to

*e*

^{iΔϕ(ρ)}from 1,

*the scattered wave is*[

*e*

^{iΔϕ(ρ)}– 1]

*whereas*

*e*

^{iΔϕ(ρ)}

*is the total wave*on that plane. The probability density for light scattering into direction

*q*=

*ks*

_{⊥}, hence, is given by where One could follow the procedure outlined in Ref [2,3

**36**(7), 1215–1217 (2011). [CrossRef] [PubMed]

*g*–

*ϕ*relation (9) if the correct probability density Eq. (16) for light scattering into direction

*q*is used.

## 5. Conclusion

*μ*–

_{s}*ϕ*,

*μ*′

*–*

_{s}*ϕ*, and

*g*–

*ϕ*relations have been provided, for the first time, with relaxed requirement on the thickness of the slice. The condition for the scattering-phase theorem to be valid has been discussed and illustrated with simulated data. The scattering-phase theorem has been applied to determine successfully the scattering coefficient, the reduced scattering coefficient and the anisotropy factor for polystyrene sphere and Intralipid-20% suspensions from their respective quantitative phase map of a thin slice.

*μ*,

_{s}*μ*′

*, and*

_{s}*g*) of biological tissue and cells has been a challenging and important problem in biomedical optics [30

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

*μ*,

_{s}*μ*′

*and*

_{s}*g*maps obtained via such a scattering-phase relationship will provide detailed local maps for scattering structures which may be of important diagnosis value, and may find applications in the characterization of the optical property of homogeneous and heterogeneous random media in general.

## Acknowledgments

## References and links

1. | W. F. Cheong, S. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. |

2. | 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). |

3. | Z. Wang, H. Ding, and G. Popescu, “Scattering-phase theorem.” Opt. Lett. |

4. | 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 |

5. | A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy.” Opt. Lett. |

6. | 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. |

7. | G. Popescu, T. Ikeda, R. R. Dasari, and M. S. Feld, “Diffraction phase microscopy for quantifying cell structure and dynamics,” Opt. Lett. |

8. | W. S. Rockward, A. L. Thomas, B. Zhao, and C. A. DiMarzio, “Quantitative phase measurements using optical quadrature microscopy,” Appl. Opt. |

9. | 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. |

10. | H. Ding, Z. Wang, F. Nguyen, S. A. Boppart, and G. Popescu, “Fourier Transform Light Scattering of Inhomogeneous and Dynamic Structures,” Phys. Rev. Lett. |

11. | S. H. Ma, |

12. | H. C. van de Hulst, |

13. | The factor |

14. | S. A. Ackerman and G. L. Stephens, “The absorption of solar radiation by cloud droplets: an application of anomalous diffraction theory,” J. Atmos. Sci. |

15. | P. Chýlek and J. D. Klett, “Extinction cross sections of nonspherical particles in the anomalous diffraction approximation,” J. Opt. Soc. Am. A |

16. | M. Xu, M. Lax, and R. R. Alfano, “Light anomalous diffraction using geometrical path statistics of rays and Gaussian ray approximation,” Opt. Lett. |

17. | 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. |

18. | M. Xu and A. Katz, |

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

20. | J. M. Schmitt and G. Kumar, “Turbulent nature of refractive-index variations in biological tissue,” Opt. Lett. |

21. | V. Turzhitsky, A. Radosevich, J. D. Rogers, A. Taflove, and V. Backman, “A predictive model of backscattering at subdiffusion length scales,” Biomed. Opt. Express |

22. | 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. |

23. | T. T. Wu, J. Y. Qu, and M. Xu, “Unified Mie and fractal scattering by biological cells and subcellular structures,” Opt. Lett. |

24. | M. Xu and R. R. Alfano, “Fractal mechanisms of light scattering in biological tissue and cells,” Opt. Lett. |

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

26. | W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. |

27. | H. J. van Staveren, C. J. M. Moes, J. van Marle, S. A. Prahl, and M. J. C. van GemertJ, “Light scattering in Intralipid-10% in the wavelength range of 400–1100nm,” Appl. Opt. |

28. | A. Giusto, R. Saija, M. A. Iat, P. Denti, F. Borghese, and O. I. Sindoni, “Optical properties of high-density dispersions of particles: application to intralipid solutions,” Appl. Opt. |

29. | 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 |

30. | S. Menon, Q. Su, and R. Grobe, “Determination of |

**OCIS Codes**

(180.3170) Microscopy : Interference microscopy

(290.5820) Scattering : Scattering measurements

(290.7050) Scattering : Turbid media

(290.5825) Scattering : Scattering theory

**ToC Category:**

Scattering

**History**

Original Manuscript: July 28, 2011

Revised Manuscript: September 16, 2011

Manuscript Accepted: September 21, 2011

Published: October 19, 2011

**Virtual Issues**

Vol. 6, Iss. 11 *Virtual Journal for Biomedical Optics*

**Citation**

Min Xu, "Scattering-phase theorem: anomalous diffraction by forward-peaked scattering media," Opt. Express **19**, 21643-21651 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-22-21643

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### References

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