## Lateral light scattering in fibrous media |

Optics Express, Vol. 21, Issue 6, pp. 7835-7840 (2013)

http://dx.doi.org/10.1364/OE.21.007835

Acrobat PDF (777 KB)

### Abstract

Lateral light scattering in fibrous media is investigated by computing the modulation transfer function (MTF) of 22 paper samples using a Monte Carlo model. The simulation tool uses phase functions from infinitely long homogenous cylinders and the directional inhomogeneity of paper is achieved by aligning the cylinders in the plane. The inverse frequency at half maximum of the MTF is compared to both measurements and previous simulations with isotropic and strongly forward single scattering phase functions. It is found that the conical scattering by cylinders enhances the lateral scattering and therefore predicts a larger extent of lateral light scattering than models using rotationally invariant single scattering phase functions. However, it does not fully reach the levels of lateral scattering observed in measurements. It is argued that the hollow lumen of a wood fiber or dependent scattering effects must be considered for a complete description of lateral light scattering in paper.

© 2013 OSA

## 1. Introduction

2. T. Linder and T. Löfqvist, “Anisotropic light propagation in paper,” Nord. Pulp Pap. Res. J. **27**, 500–506 (2012) [CrossRef] .

3. M. Neuman, L. G. Coppel, and P. Edstrom, “Point spreading in turbid media with anisotropic single scattering,” Opt. Express **19**, 1915–1920 (2011) [CrossRef] [PubMed] .

*k*, with a model based on the Kubelka-Munk (KM) scattering coefficient

_{p}*S*. Recently Coppel et al. [6

6. L. G. Coppel, M. Neuman, and P. Edström, “Lateral light scattering in paper - MTF simulation and measurement,” Opt. Express **19**, 25181–25187 (2011) [CrossRef] .

8. A. Kienle, F. K. Forster, R. Diebolder, and R. Hibst, “Light propagation in dentin: influence of microstructure on anisotropy,” Phys. Med. Biol. **48**, N7–N14 (2003) [CrossRef] [PubMed] .

9. A. Kienle, F. K. Forster, and R. Hibst, “Anisotropy of light propagation in biological tissue,” Opt. Lett. **29**, 2617–2619 (2004) [CrossRef] [PubMed] .

10. H. He, N. Zeng, R. Liao, T. Yun, W. Li, Y. He, and H. Ma, “Application of sphere-cylinder scattering model to skeletal muscle,” Opt. Express **18**, 15104–15112 (2010) [CrossRef] [PubMed] .

11. A. Kienle, C. Wetzel, A. Bassi, D. Comelli, P. Taroni, and A. Pifferi, “Determination of the optical properties of anisotropic biological media using an isotropic diffusion model,” J. Biomed. Opt. **12** (2007) [CrossRef] [PubMed] .

12. A. Kienle, C. D’Andrea, F. Foschum, P. Taroni, and A. Pifferi, “Light propagation in dry and wet softwood,” Opt. Express **16**, 9895–9906 (2008) [CrossRef] [PubMed] .

13. B. Peng, T. Ding, and P. Wang, “Propagation of polarized light through textile material,” Appl. Opt. **51**, 6325–6334 (2012) [CrossRef] [PubMed] .

14. T. Yun, N. Zeng, W. Li, D. Li, X. Jiang, and H. Ma, “Monte Carlo simulation of polarized photon scattering in anisotropic media,” Opt. Express **17**, 16590–16602 (2009) [CrossRef] [PubMed] .

*g*is no longer required in the same way as in for example the Henyey-Greenstein phase function. Another benefit is that the scattering efficiency

*Q*can be obtained, it can be used to define how scattering distance depend on the absolute direction of photons propagating in aligned fiber structures.

_{s}6. L. G. Coppel, M. Neuman, and P. Edström, “Lateral light scattering in paper - MTF simulation and measurement,” Opt. Express **19**, 25181–25187 (2011) [CrossRef] .

*k*to test whether they can predict the amount of lateral scattering in the real paper samples.

_{p}## 2. Method

*g*= 0.0) and forward single scattering (

*g*= 0.8) and the simulated values of

*k*are also compared to the measured values.

_{p}### 2.1. Material parameters and parameter estimation

*σ*= 5° in the thickness direction. A more detailed description of the cylinder alignment has been reported in [2

_{z}2. T. Linder and T. Löfqvist, “Anisotropic light propagation in paper,” Nord. Pulp Pap. Res. J. **27**, 500–506 (2012) [CrossRef] .

*n*= 1.55) is used for the cylinders and the surrounding medium is air (

*n*= 1.0). The cylinder diameter is chosen to be 20

*μ*m as typical wood fiber dimension [15]. The wavelength of the light was chosen to be

*λ*= 510 nm as the experiments in Arney et al. used a green filter. We optimized

*C*iteratively using a Gauss-Newton method in the 0°/

_{a}*d*geometry with the Monte Carlo simulation tool to match the measured total transmission and reflection for each of the paper samples with their given thicknesses. The obtained cylinder density

*C*, scattering coefficient in the thickness direction

_{a}*μ*(

_{s}**z̄**) and absorption coefficient

*μ*are shown in Table 1.

_{a}### 2.2. Simulation of the edge response

*http://fibermc.sourceforge.net/*) is used to compute the point spread function (PSF) for each of the 22 sets of paper sheets. Both for a single paper sheet and for an opaque pad of paper sheets. The resolution was chosen to 10

*μ*m and the incident trajectories of the photons was tilted with a 20° angle toward the surface normal to match the measurements. A 2D convolution between the simulated point spread function and a intensity distribution

*i*(

*x*,

*y*) was used to derive the edge response The intensity distribution is uniformly distributed and cut off by a sharp knife edge, i.e. a matrix consisting of 1’s on one side and 0’s on the other. A similar convolution was used by Ukishima et al. [16] who considered the intensity distribution of pencil light together with the point spread function. This approach is different compared to Coppel et al. who simulated the edge response directly by distributing each of the photons over an area. Since the statistical response of a PSF is more focused it greatly reduces the amount of simulated photons required to keep down the noise levels. It is therefore preferred to use the convolution as Monte Carlo simulations are very time consuming.

### 2.3. MTF

*k*, is defined as Only measured values of

_{p}*k*are compared to the simulations since Arney et al. only reported

_{p}*k*. The full MTF would have been preferred over the metric

_{p}*k*as it only indicates the amount of lateral light scattering and lose a lot of the information held by the MTF [6

_{p}6. L. G. Coppel, M. Neuman, and P. Edström, “Lateral light scattering in paper - MTF simulation and measurement,” Opt. Express **19**, 25181–25187 (2011) [CrossRef] .

## 3. Results

*g*= 0.8) at higher frequencies. The measured values of 1/

*k*are presented by red dots in the figures. Figure 2 shows the simulated results for all the 22 samples for both single sheets and opaque pads of paper sheets in relation to the measurements and previous simulations. The general trend is that the simulated values of

_{p}*k*are underestimated in comparison to measurements but they predict larger values than simulations with rotationally invariant phase functions.

_{p}*k*in relation to

_{p}*C*for a single sheet and an opaque pad of paper samples. At low concentrations, or low scattering coefficients, the cylinder model seems to agree better with the previous simulations than with the measurements. It appears that the best correlation to the measurements are found in the middle region where

_{a}*C*lie between 3· 10

_{a}^{9}and 5·10

^{9}

*m*

^{−2}. For larger values of

*C*a decresing correlation can be observed.

_{a}## 4. Discussion and conclusions

*k*than models using rotationally invariant isotropic and strong forward scattering. However, the cylinder model still underestimates

_{p}*k*compared to the measured values. This means that the model predicting less lateral scattering than observed.

_{p}*d*and wavelength

*λ*. We observed that this did not have any significant effect on the shape of the PSF and the resulting MTFs. This indicates that the model cannot predict more lateral light scattering for parameters within the natural parameter range of wood fibers. Arney suggested that the hollow lumen of a wood fiber could increase the lateral light scattering through a light-piping effect. This seem to be a reasonable explanation to why the model still predicts less lateral scattering than measured. Another reasonable explanation is that the concentration of particles is high, causing dependent scattering effects giving rise to interactions which are different from the interdependent scattering theory used in this work. Part of the lateral scattering can, however, be explained by the conical scattering by cylindrical objects. Compared to the Monte Carlo simulations using the Henyey-Greenstain phase function it roughly closes half the gap between earlier MC simulations and the experimental results. The cylinder phase function gives a good idea of how the lateral light scattering in fibrous materials is generated and has potential to better model for example optical dot gain.

## References and links

1. | J. A. C. Yule and W. J. Nielsen, “The penetration of light into paper and its effect on halftone reproduction,” Proceedings of TAGA |

2. | T. Linder and T. Löfqvist, “Anisotropic light propagation in paper,” Nord. Pulp Pap. Res. J. |

3. | M. Neuman, L. G. Coppel, and P. Edstrom, “Point spreading in turbid media with anisotropic single scattering,” Opt. Express |

4. | J. Arney, C. Arney, M. Katsube, and P. Engeldrum, “An MTF analysis of papers,” J. Imaging Sci. Technol. |

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

6. | L. G. Coppel, M. Neuman, and P. Edström, “Lateral light scattering in paper - MTF simulation and measurement,” Opt. Express |

7. | C. F. Bohren and D. R. Huffman, |

8. | A. Kienle, F. K. Forster, R. Diebolder, and R. Hibst, “Light propagation in dentin: influence of microstructure on anisotropy,” Phys. Med. Biol. |

9. | A. Kienle, F. K. Forster, and R. Hibst, “Anisotropy of light propagation in biological tissue,” Opt. Lett. |

10. | H. He, N. Zeng, R. Liao, T. Yun, W. Li, Y. He, and H. Ma, “Application of sphere-cylinder scattering model to skeletal muscle,” Opt. Express |

11. | A. Kienle, C. Wetzel, A. Bassi, D. Comelli, P. Taroni, and A. Pifferi, “Determination of the optical properties of anisotropic biological media using an isotropic diffusion model,” J. Biomed. Opt. |

12. | A. Kienle, C. D’Andrea, F. Foschum, P. Taroni, and A. Pifferi, “Light propagation in dry and wet softwood,” Opt. Express |

13. | B. Peng, T. Ding, and P. Wang, “Propagation of polarized light through textile material,” Appl. Opt. |

14. | T. Yun, N. Zeng, W. Li, D. Li, X. Jiang, and H. Ma, “Monte Carlo simulation of polarized photon scattering in anisotropic media,” Opt. Express |

15. | C. Fellers and B. Norman, |

16. | M. Ukishima, H. Kaneko, T. Nakaguchi, N. Tsumura, M. Hauta-Kasari, J. Parkkinen, and Y. Miyake, “A Simple Method to Measure MTF of Paper and Its Application for Dot Gain Analysis,” IEICE Trans. Fundam. Electron. Commun. Comput. Sci. |

**OCIS Codes**

(100.2810) Image processing : Halftone image reproduction

(290.4210) Scattering : Multiple scattering

(290.7050) Scattering : Turbid media

(010.5620) Atmospheric and oceanic optics : Radiative transfer

**ToC Category:**

Scattering

**History**

Original Manuscript: February 25, 2013

Revised Manuscript: March 15, 2013

Manuscript Accepted: March 16, 2013

Published: March 22, 2013

**Virtual Issues**

Vol. 8, Iss. 4 *Virtual Journal for Biomedical Optics*

**Citation**

Tomas Linder, Torbjörn Löfqvist, Ludovic G. Coppel, Magnus Neuman, and Per Edström, "Lateral light scattering in fibrous media," Opt. Express **21**, 7835-7840 (2013)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-21-6-7835

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### References

- J. A. C. Yule and W. J. Nielsen, “The penetration of light into paper and its effect on halftone reproduction,” Proceedings of TAGA3, 65–67 (1951).
- T. Linder and T. Löfqvist, “Anisotropic light propagation in paper,” Nord. Pulp Pap. Res. J.27, 500–506 (2012). [CrossRef]
- M. Neuman, L. G. Coppel, and P. Edstrom, “Point spreading in turbid media with anisotropic single scattering,” Opt. Express19, 1915–1920 (2011). [CrossRef] [PubMed]
- J. Arney, C. Arney, M. Katsube, and P. Engeldrum, “An MTF analysis of papers,” J. Imaging Sci. Technol.40, 19–25 (1996).
- 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).
- L. G. Coppel, M. Neuman, and P. Edström, “Lateral light scattering in paper - MTF simulation and measurement,” Opt. Express19, 25181–25187 (2011). [CrossRef]
- C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley and Sons, 1983).
- A. Kienle, F. K. Forster, R. Diebolder, and R. Hibst, “Light propagation in dentin: influence of microstructure on anisotropy,” Phys. Med. Biol.48, N7–N14 (2003). [CrossRef] [PubMed]
- A. Kienle, F. K. Forster, and R. Hibst, “Anisotropy of light propagation in biological tissue,” Opt. Lett.29, 2617–2619 (2004). [CrossRef] [PubMed]
- H. He, N. Zeng, R. Liao, T. Yun, W. Li, Y. He, and H. Ma, “Application of sphere-cylinder scattering model to skeletal muscle,” Opt. Express18, 15104–15112 (2010). [CrossRef] [PubMed]
- A. Kienle, C. Wetzel, A. Bassi, D. Comelli, P. Taroni, and A. Pifferi, “Determination of the optical properties of anisotropic biological media using an isotropic diffusion model,” J. Biomed. Opt.12 (2007). [CrossRef] [PubMed]
- A. Kienle, C. D’Andrea, F. Foschum, P. Taroni, and A. Pifferi, “Light propagation in dry and wet softwood,” Opt. Express16, 9895–9906 (2008). [CrossRef] [PubMed]
- B. Peng, T. Ding, and P. Wang, “Propagation of polarized light through textile material,” Appl. Opt.51, 6325–6334 (2012). [CrossRef] [PubMed]
- T. Yun, N. Zeng, W. Li, D. Li, X. Jiang, and H. Ma, “Monte Carlo simulation of polarized photon scattering in anisotropic media,” Opt. Express17, 16590–16602 (2009). [CrossRef] [PubMed]
- C. Fellers and B. Norman, Pappersteknik, 3rd ed. (Department of Pulp and Paper Chemistry and Technology, Royal Institute of Technology, 1996).
- M. Ukishima, H. Kaneko, T. Nakaguchi, N. Tsumura, M. Hauta-Kasari, J. Parkkinen, and Y. Miyake, “A Simple Method to Measure MTF of Paper and Its Application for Dot Gain Analysis,” IEICE Trans. Fundam. Electron. Commun. Comput. Sci.E92A, 3328–3335.

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