## Point spreading in turbid media with anisotropic single scattering |

Optics Express, Vol. 19, Issue 3, pp. 1915-1920 (2011)

http://dx.doi.org/10.1364/OE.19.001915

Acrobat PDF (646 KB)

### Abstract

Point spreading is investigated using general radiative transfer theory. We find that the single scattering anisotropy plays a significant role for point spreading together with the medium mean free path, single scattering albedo and thickness. When forward scattering dominates, the light will on average undergo more scattering events to give a specific optical response in reflectance measurements. This will significantly increase point spreading if the medium is low absorbing with large mean free path. Any fundamental and generic model of point spreading must capture the dependence on all of these medium characteristics.

© 2011 Optical Society of America

## 1. Introduction

*D*(

*x,y*) of the reflected light is obtained by integrating the PSF and the illumination

*I*over the reflecting medium, that is where Δ

*u*and Δ

*v*delimit the medium. In the present work we deal with laterally homogeneous turbid media with point source illumination. The PSF is then isotropic, i.e., it depends on radial distance only, and the reflected light is given by

*D*(

*r*) =

*PSF*(

*r*)

*I*if the illumination is incident at radial coordinate

*r*= 0.

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

*I*(

*x,y,z*,

*θ*,

*φ*) is intensity at position (

*x,y,z*) at polar angle

*θ*and azimuthal angle

*φ*,

*s*= (

*x*

^{2}+

*y*

^{2}+

*z*

^{2})

^{1/2}is distance,

*σ*is the extinction coefficient and

_{e}*S*is a source function. The extinction coefficient is the sum of the scattering and absorption coefficients

*σ*and

_{s}*σ*and also the inverse of the mean free path

_{a}*ℓ*. The source function accounts for light scattered to

_{e}*θ*,

*φ*at position (

*x,y,z*) from all other directions. It can be written where

*a*is the single scattering albedo defined as

*a*=

*σ*/(

_{s}*σ*+

_{s}*σ*),

_{a}*ω*is solid angle and

*p*(cos Θ) is the phase function. Here Θ is the angle between the directions of the incident and scattered light. The phase function describes the angular distribution of the single scattering process. A commonly used phase function is the Henyey–Greenstein (HG) phase function [11

11. L. G. Henyey and J. L. Greenstein, “Diffuse Radiation in the Galaxy,” Astrophys. J. **93**, 70–83 (1941). [CrossRef]

*g*. It ranges from −1 to 1 with

*g*= −1 meaning complete back scattering,

*g*= 0 isotropic scattering and

*g*= 1 complete forward scattering. Also,

*g*is the average of the cosine of the scattering angle and the first moment in an expansion of any phase function. For organic materials such as paper it has been shown that

*g*has values in the interval 0.6–0.9 approximately [12

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

14. N. Joshi, C. Donner, and H. W. Jensen, “Noninvasive measurement of scattering anisotropy in turbid materials by nonnormal incident illumination,” Opt. Lett. **31**, 936–938 (2006). [CrossRef] [PubMed]

*a*, the mean free path

*ℓ*and the asymmetry factor

_{e}*g*. These medium characteristics are independent of position in a homogeneous medium but normally vary between wavelengths. Equation (2) can then be solved for each wavelength if the wavelengths are independent. This is not the case in fluorescing media, but in the present work we do not consider fluorescence.

15. P. Edström, “A fast and stable solution method for the radiative transfer problem,” SIAM Rev. **47**, 447–468 (2005). [CrossRef]

17. M. Sormaz, T. Stamm, S. Mourad, and P. Jenny, “Stochastic modeling of light scattering with fluorescence using a Monte Carlo-based multiscale approach,” J. Opt. Soc. Am. A **26**, 1403–1413 (2009). [CrossRef]

18. T. F. Chen, G. V. G. Baranoski, and K. F. Lin, “Bulk scattering approximations for HeNe laser transmitted through paper,” Opt. Express **16**, 21762–21771 (2008). [CrossRef] [PubMed]

19. M. Neuman and P. Edström, “Anisotropic reflectance from turbid media. I. Theory,” J. Opt. Soc. Am. A **27**, 1032–1039 (2010). [CrossRef]

20. M. Neuman and P. Edström, “Anisotropic reflectance from turbid media. II. Measurements,” J. Opt. Soc. Am. A **27**, 1040–1045 (2010). [CrossRef]

## 2. Method

### 2.1. Material

*ℓ*and

_{e}*a*are assessed using a set of paper samples. We choose two lightly dyed and two non-dyed 30 g/m

^{2}paper samples with and without fillers, giving four samples in total. Samples varying in filler content differ in mean free path

*l*since the addition of fillers significantly increases scattering, thus decreasing the mean free path. The addition of a blue dye increases absorption, thus decreasing the albedo, in the wavelength interval 550–700 nm approximately, and the effect of increased absorption on point spreading can be investigated by studying wavelengths in this interval. No samples contain fluorescent whitening agents. We denote the samples M1–M4, where M1 contains no dye or filler, M2 contains dye but no filler, M3 contains no dye but filler and M4 contains both dye and filler.

_{e}### 2.2. Estimation of medium parameters using DORT simulations

*ℓ*and

_{e}*a*by measuring the reflectance factor from a single sheet and an opaque pad of paper sheets. We then get a well-posed optimization problem that can be solved for

*ℓ*and

_{e}*a*, e.g., by using the RT based DORT2002 model [15

15. P. Edström, “A fast and stable solution method for the radiative transfer problem,” SIAM Rev. **47**, 447–468 (2005). [CrossRef]

*g*is varied from 0 to 0.8 in steps of 0.2, and the inverse RT problem is solved for each of these

*g*values. We include

*g*= 0 since this is an assumption in the KM model. Each parameter setup [

*a, ℓ*,

_{e}*g*] will then give the same optical response in the d/0 instrument for the particular medium studied, despite the variations in

*g*, and the medium thus has the same scattering power irrespective of the

*g*value.

### 2.3. Monte Carlo simulations of the PSF

*a, ℓ*,

_{e}*g*] we can estimate the PSF of a medium with a given thickness

*t*through Monte Carlo simulations. We do simulations using both the thickness

*t*corresponding to 30 g/m

^{2}paper and an opaque medium where

*t*→ ∞ in order to study the influence of transmittance on the PSF. The thickness of the paper samples is measured with a micrometer and found to be 65

*μ*m. We simulate illumination incident normally on a point and to minimize noise 10

^{8}wave packets are simulated in each run. We choose the wavelength of light that is most heavily absorbed by the blue dye (620 nm) to represent the four media M1–4.

## 3. Results

### 3.1. Medium parameters

*a*and

*ℓ*obtained from d/0 measurements when

_{e}*g*is varied. As expected M2 and M4 have the lowest albedos and M3 and M4 have the shortest mean free path. We can see that the mean free path decreases as

*g*increases. This can be understood intuitively since if the light is scattered more in the forward direction when it impinges on the medium surface, the medium must be highly scattering in order to reflect the measured amount of light towards the detector. In this way the scattering power is the same irrespective of the

*g*value.

### 3.2. Point spread simulations

*r̄*varies with

*g*for the different media and for the two different thicknesses. It can be seen that

*r̄*increases with

*g*for all media. The high albedo medium with large mean free path (M1) has the largest mean radial distance

*r̄*. The low albedo medium with short mean free path (M4) has the smallest

*r̄*. We can also see that

*r̄*is larger for the opaque media, with the most noticeable difference for M1. Hence, transmittance can obviously have a significant effect on point spreading. We can thus conclude that point spreading, as represented by

*r̄*, depends on asymmetry factor, albedo, mean free path and medium thickness.

*g*increases the number of scattering events. This holds for both thin and opaque media, but when the mean free path

*ℓ*is large (as for M1 and M2) the light is scattered fewer times in thin media. When the effect of transmittance is eliminated (Fig. 2(b)) there is only a small difference in the number of scattering events when varying the mean free path (M1 vs. M3 and M2 vs. M4).

_{e}## 4. Discussion and conclusions

## Acknowledgments

## References and links

1. | P. Oittinen, “Limits of microscopic print quality,” in |

2. | P. Kubelka and F. Munk, “Ein beitrag zur optik der farbanstriche,” Z. Tech. Phys. (Leipzig) |

3. | P. G. Engeldrum and B. Pridham, “Application of turbid medium theory to paper spread function measurements,” Tech. Assoc. Graphic Arts Proc. |

4. | J.S. Arney, J. Chauvin, J. Nauman, and P.G. Anderson, “Kubelka-Munk theory and the MTF of paper,” J. Imaging Sci. Technol. |

5. | S. Gustavson, “Dot Gain in Colour Halftones,” Doctoral thesis, Linköping university (1997). |

6. | P. Emmel, “Modèles de Prédiction Couleur Appliqués à l’Impression Jet d’Encre,” Doctoral thesis, Ecole Polytechnique Fédérale de Lausanne (1998). |

7. | S. Mourad, “Improved Calibration of Optical Characteristics of Paper by an Adapted Paper-MTF Model,” J. Imaging Sci. Technol. |

8. | A. S. Glassner, |

9. | J. M. Schmitt, “Optical coherence tomography (OCT): A review,” IEEE J. Sel. Top. Quantum Electron. |

10. | S. Chandrasekhar, |

11. | L. G. Henyey and J. L. Greenstein, “Diffuse Radiation in the Galaxy,” Astrophys. J. |

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

13. | S. A. Prahl, M. J. C. van Gemert, and A. J. Welch, “Determining the optical properties of turbid media by using the adding-doubling method,” Appl. Opt. |

14. | N. Joshi, C. Donner, and H. W. Jensen, “Noninvasive measurement of scattering anisotropy in turbid materials by nonnormal incident illumination,” Opt. Lett. |

15. | P. Edström, “A fast and stable solution method for the radiative transfer problem,” SIAM Rev. |

16. | L. G. Coppel, P. Edström, and M. Lindquister, “Open source Monte Carlo simulation platform for particle level simulation of light scattering from generated paper structures,” in |

17. | M. Sormaz, T. Stamm, S. Mourad, and P. Jenny, “Stochastic modeling of light scattering with fluorescence using a Monte Carlo-based multiscale approach,” J. Opt. Soc. Am. A |

18. | T. F. Chen, G. V. G. Baranoski, and K. F. Lin, “Bulk scattering approximations for HeNe laser transmitted through paper,” Opt. Express |

19. | M. Neuman and P. Edström, “Anisotropic reflectance from turbid media. I. Theory,” J. Opt. Soc. Am. A |

20. | M. Neuman and P. Edström, “Anisotropic reflectance from turbid media. II. Measurements,” J. Opt. Soc. Am. A |

21. | ISO 2469: |

22. | P. Edström, “A Two-Phase Parameter Estimation Method for Radiative Transfer Problems in Paper Industry Applications,” J. Comput. Appl. Math. |

**OCIS Codes**

(030.5620) Coherence and statistical optics : Radiative transfer

(100.2810) Image processing : Halftone image reproduction

(170.3660) Medical optics and biotechnology : Light propagation in tissues

(290.4210) Scattering : Multiple scattering

(290.7050) Scattering : Turbid media

(290.2558) Scattering : Forward scattering

**ToC Category:**

Scattering

**History**

Original Manuscript: November 18, 2010

Revised Manuscript: January 12, 2011

Manuscript Accepted: January 12, 2011

Published: January 18, 2011

**Virtual Issues**

Vol. 6, Iss. 2 *Virtual Journal for Biomedical Optics*

**Citation**

Magnus Neuman, Ludovic G. Coppel, and Per Edström, "Point spreading in turbid media with anisotropic single scattering," Opt. Express **19**, 1915-1920 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-3-1915

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### References

- P. Oittinen, "Limits of microscopic print quality," in Advances in Printing Science and Technology, W. H. Banks, ed. (Pentech, London, 1982), Vol. 16, pp 121-128.
- P. Kubelka, and F. Munk, "Ein beitrag zur optik der farbanstriche," Z. Tech. Phys. (Leipzig) 11a, 593-601 (1931).
- P. G. Engeldrum, and B. Pridham, "Application of turbid medium theory to paper spread function measurements," Tech. Assoc. Graphic Arts Proc. 47, 339-352 (1995).
- J. S. Arney, J. Chauvin, J. Nauman, and P. G. Anderson, "Kubelka-Munk theory and the MTF of paper," J. Imaging Sci. Technol. 47, 339-345 (2003).
- S. Gustavson, "Dot Gain in Colour Halftones," Doctoral thesis, Linköping university (1997).
- P. Emmel, "Modèles de Prédiction Couleur Appliqués à l’Impression Jet d’Encre," Doctoral thesis, Ecole Polytechnique Fédérale de Lausanne (1998).
- S. Mourad, "Improved Calibration of Optical Characteristics of Paper by an Adapted Paper-MTF Model," J. Imaging Sci. Technol. 51, 283-292 (2007). [CrossRef]
- A. S. Glassner, Principles of Digital Image Synthesis, Vol. 2, (Morgan Kauffman, 1995).
- J. M. Schmitt, "Optical coherence tomography (OCT): A review," IEEE J. Sel. Top. Quantum Electron. 5, 1205-1215 (1999). [CrossRef]
- S. Chandrasekhar, Radiative Transfer, (Dover, 1960).
- L. G. Henyey, and J. L. Greenstein, "Diffuse Radiation in the Galaxy," Astrophys. J. 93, 70-83 (1941). [CrossRef]
- W.-F. Cheong, S. A. Prahl, and A. J. Welch, "A review of the optical properties of biological tissues," IEEE J. Quantum Electron. 26, 2166-2185 (1990). [CrossRef]
- S. A. Prahl, M. J. C. van Gemert, and A. J. Welch, "Determining the optical properties of turbid media by using the adding-doubling method," Appl. Opt. 32, 559-568 (1993). [CrossRef] [PubMed]
- N. Joshi, C. Donner, and H. W. Jensen, "Noninvasive measurement of scattering anisotropy in turbid materials by nonnormal incident illumination," Opt. Lett. 31, 936-938 (2006). [CrossRef] [PubMed]
- P. Edström, "A fast and stable solution method for the radiative transfer problem," SIAM Rev. 47, 447-468 (2005). [CrossRef]
- L. G. Coppel, P. Edström, and M. Lindquister, "Open source Monte Carlo simulation platform for particle level simulation of light scattering from generated paper structures," in Proc. Papermaking Res. Symp., E. Madetoja, H. Niskanen and J. Hämäläinen, eds. (Kuopio, 2009).
- M. Sormaz, T. Stamm, S. Mourad, and P. Jenny, "Stochastic modeling of light scattering with fluorescence using a Monte Carlo-based multiscale approach," J. Opt. Soc. Am. A 26, 1403-1413 (2009). [CrossRef]
- T. F. Chen, G. V. G. Baranoski, and K. F. Lin, "Bulk scattering approximations for HeNe laser transmitted through paper," Opt. Express 16, 21762-21771 (2008). [CrossRef] [PubMed]
- M. Neuman, and P. Edström, "Anisotropic reflectance from turbid media. I. Theory," J. Opt. Soc. Am. A 27, 1032-1039 (2010). [CrossRef]
- M. Neuman, and P. Edström, "Anisotropic reflectance from turbid media. II. Measurements," J. Opt. Soc. Am. A 27, 1040-1045 (2010). [CrossRef]
- ISO 2469: Paper, Board and Pulps - Measurement of Diffuse Reflectance Factor, (International Organization for Standardization, 1994).
- P. Edström, "A Two-Phase Parameter Estimation Method for Radiative Transfer Problems in Paper Industry Applications," J. Comput. Appl. Math. 16, 927-951 (2008).

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