## Photon transport in cylindrically-shaped disordered meso-macroporous materials |

Optics Express, Vol. 22, Issue 7, pp. 7503-7513 (2014)

http://dx.doi.org/10.1364/OE.22.007503

Acrobat PDF (2682 KB)

### Abstract

We theoretically and experimentally investigate light diffusion in disordered meso-macroporous materials with a cylindrical shape. High Internal Phase Emulsion (HIPE)-based silica foam samples, exhibiting a polydisperse pore-size distribution centered around 19 *μ*m to resemble certain biological tissues, are realized. To quantify the effect of a finite lateral size on measurable quantities, an analytical model for diffusion in finite cylinders is developed and validated by Monte Carlo random walk simulations. Steady-state and time-resolved transmission experiments are performed and the transport parameters (transport mean free path and material absorption length) are successfully retrieved from fits of the experimental curves with the proposed model. This study reveals that scattering losses on the lateral sides of the samples are responsible for a lowering of the transmission signal and a shortening of the photon lifetime, similar in experimental observables to the effect of material absorption. The recognition of this geometrical effect is essential since its wrong attribution to material absorption could be detrimental in various applications, such as biological tissue diagnosis or conversion efficiency in dye-sensitized solar cells.

© 2014 Optical Society of America

## 1. Introduction

1. A. B. Davis and A. Marshak, “Solar radiation transport in the cloudy atmosphere: a 3D perspective on observations and climate impacts,” Rep. Prog. Phys. **73**, 026801 (2010). [CrossRef]

3. G. Reich, “Near-infrared spectroscopy and imaging: Basic principles and pharmaceutical applications,” Adv. Drug Delivery Rev. **57**, 1109 (2005). [CrossRef]

5. V. V. Tuchin, *Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis*, 2 (SPIE, Bellingham, WA, 2007). [CrossRef]

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

9. T. Svensson, E. Alerstam, D. Khoptyar, J. Johansson, S. Folestad, and S. Andersson-Engels, “Near-infrared photon time-of-flight spectroscopy of turbid materials up to 1400 nm,” Rev. Sci. Instrum. **80**, 063105 (2009). [CrossRef] [PubMed]

10. V. Backman, M. B. Wallace, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Müller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, S. Shapshay, T. Valdez, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, J. Van Dam, and M. S. Feld, “Detection of preinvasive cancer cells,” Nature **406**, 35–36 (2000). [CrossRef] [PubMed]

11. E. Alerstam and T. Svensson, “Observation of anisotropic diffusion of light in compacted granular porous materials,” Phys. Rev. E **89**, 040301 (2012). [CrossRef]

12. T. Svensson, M. Andersson, L. Rippe, S. Svanberg, S. Andersson-Engels, J. Johansson, and S. Folestad, “VCSEL-based oxygen spectroscopy for structural analysis of pharmaceutical solids,” Appl. Phys. B **90**, 345–354 (2008). [CrossRef]

13. Z. Shi and C. A. Anderson, “Pharmaceutical applications of separation of absorption and scattering in near-infrared spectroscopy (NIRS),” J. Pharm. Sci. **99**, 4766–4783 (2010). [CrossRef] [PubMed]

14. C. M. Leroy, C. Olivier, T. Toupance, M. Abbas, L. Hirsch, S. Ravaine, and R. Backov, “One-pot easily-processed TiO_{2} macroporous photoanodes (Ti-HIPE) for dye-sensitized solar cells,” Sol. State Sci. **28**, 81–89 (2014). [CrossRef]

## 2. Material fabrication and structural characterization

26. F. Carn, A. Colin, M-.F. Achard, M. Pirot, H. Deleuze, and R. Backov, “Inorganic monoliths hierarchically textured via concentrated direct emulsion and micellar templates,” J. Mat. Chem. **14**, 1370–1376 (2004). [CrossRef]

*f*and the shearing rate.

_{4}, TEOS) and tetradecyltrimethylammonium bromide 98% (C

_{14}H

_{29}NBr(CH

_{3}), TTAB) were purchased from Fluka, HCl 37% and dodecane 99% were purchased from Prolabo. Procedures are based on the use of both micelles and direct concentrated emulsion templates [26

26. F. Carn, A. Colin, M-.F. Achard, M. Pirot, H. Deleuze, and R. Backov, “Inorganic monoliths hierarchically textured via concentrated direct emulsion and micellar templates,” J. Mat. Chem. **14**, 1370–1376 (2004). [CrossRef]

_{2}(HIPE). These macrocellular foams emerge from the use of a starting concentrated direct emulsion at oil volumic fraction (

*f*= 0.67), where 35 g of dodecane are emulsified.

*f*= 0.67, the macrocellular cell sizes are in the range 5–50

*μ*m, with a mean value of 19

*μ*m and a standard deviation of 9.6

*μ*m (Fig. 1(b)). Such a foam closely resembles a population of epithelial cells [10

10. V. Backman, M. B. Wallace, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Müller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, S. Shapshay, T. Valdez, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, J. Van Dam, and M. S. Feld, “Detection of preinvasive cancer cells,” Nature **406**, 35–36 (2000). [CrossRef] [PubMed]

*f*= 0.74 [28

28. M. J. Mooney, “The viscosity of a concentrated suspension of spherical particles,” J. Colloid. Interface Sci. **6**, 162–170 (1951). [CrossRef]

30. M.-P. Aronson and M.-F. Petko, “Highly Concentrated Water-in-Oil Emulsions: Influence of Electrolyte on Their Properties and Stability,” J. Colloid Interface Sci. **159**, 134–149 (1993). [CrossRef]

_{2}(HIPE) materials considered here, we do not reach this limit. The macroporous monolithic texture resembles aggregated polydisperse hollow spheres. In such low pH conditions, the polycondensation is strongly Euclidian [31] (dense) and starts at the oil/water interface. The latter is promoting silica condensation by minimizing the nucleation enthalpy, acting so as a defect [27]. As the oil-water interface of an emulsion is associated to a higher surfactant concentration than the core of the continuous aqueous phase [28

28. M. J. Mooney, “The viscosity of a concentrated suspension of spherical particles,” J. Colloid. Interface Sci. **6**, 162–170 (1951). [CrossRef]

30. M.-P. Aronson and M.-F. Petko, “Highly Concentrated Water-in-Oil Emulsions: Influence of Electrolyte on Their Properties and Stability,” J. Colloid Interface Sci. **159**, 134–149 (1993). [CrossRef]

26. F. Carn, A. Colin, M-.F. Achard, M. Pirot, H. Deleuze, and R. Backov, “Inorganic monoliths hierarchically textured via concentrated direct emulsion and micellar templates,” J. Mat. Chem. **14**, 1370–1376 (2004). [CrossRef]

## 3. Theory and simulations

### 3.1. Model of diffusion in finite cylinders

*u*(

**r**,

*t*) where

*D*=

*vℓ*/3 is the diffusion constant for a statistically homogeneous and isotropic disordered medium,

_{t}*v*is the energy velocity in the medium,

*ℓ*is the transport mean free path and

_{t}*μ*= 1/

_{a}*ℓ*is the absorption coefficient of the material (

_{a}*ℓ*being the material absorption length). Considering invariance by rotation around the cylinder axis (say along

_{a}*z*), the Laplacian operator can be written in cylindrical coordinates as Performing the standard approach of separation of variables

*u*(

*ρ*,

*z*,

*t*) =

*P*(

*ρ*)

*Z*(

*z*)

*T*(

*t*), the diffusion equation leads to a system of three differential equations which can be solved directly to give Here,

*C*

_{1},

*C*

_{2},

*C*

_{3}and

*C*

_{4}are constants,

*J*

_{0}and

*Y*

_{0}are zeroth-order Bessel functions of the first and second kinds, respectively, and

*λ*and

_{z}*λ*are the eigenvalues of the differential operators in

_{ρ}*z*and

*ρ*, respectively.

*u*(

*ρ*,

*z*,

*t*). The physical dimensions of the cylindrical system are defined by its radius

*R*and its length

*L*, see Fig. 2(a). Due to the outgoing flux, the energy density in the medium does not equal zero at the physical boundaries but at a distance

*r*is the internal reflection coefficient and

*ℓ*the transport mean free path [32

_{t}32. J. X. Zhu, D. J. Pine, and D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A **44**, 3948–3959 (1991). [CrossRef] [PubMed]

*R*=

_{e}*R*+

*l*and length

_{e}*L*=

_{e}*L*+ 2

*l*. The boundary conditions are the following. From these conditions, we find a discrete set of eigenvalues,

_{e}*λ*

_{ρ}_{,}

*= (*

_{m}*α*/

_{m}*R*)

_{e}^{2}

*D*and

*λ*

_{z}_{,}

*= (*

_{n}*nπ*/

*L*)

_{e}^{2}

*D*, where

*α*is the

_{m}*m*th zero of

*J*

_{0}. By applying the orthogonality relation between eigenmodes over the extrapolated volume of the system, we obtain an expression for the energy density:

*t*= 0) to be a disk of radius

*l*at a depth

_{ε}*z*=

*z*

_{0}, i.e.

*l*→ 0 to have a point source.

_{ε}*z*-axis, such that we can limit ourselves to the calculation of

*u*(

*ρ*,

*z*,

*t*) with

*z*is straightforward and leads to

*z*-axis, we should integrate

*J*over the surface, as

_{z}*z*-axis of the cylinder, it is sufficient to integrate

*T*(

*z*,

*t*) over time as

*R*→ ∞, Eqs. (13) and (14) converge correctly to the solutions for slab geometries. Equations (13) and (14) hold for all values of

*R*and

*L*and therefore generalize the standard expressions for slab geometries.

### 3.2. Comparison with Monte Carlo simulations

33. L.-H. Wang, S. L. Jacques, and L.-Q. Zheng, ”Monte Carlo modeling of photon transport in multi-layered tissues,” Computer Methods and Programs in Biomedicine **47**, 131–146 (1995). [CrossRef]

34. M. Xu and R. R. Alfano, “Random walk of polarized light in turbid media,” Phys. Rev. Lett. **95**, 213901 (2005). [CrossRef] [PubMed]

*μ*= 0) medium with a refractive index of 1 such that internal reflections can be neglected (

_{a}*r*= 0) and

*v*= 300

*μ*m/ps. We recall that

*R*=

_{e}*R*+

*l*,

_{e}*L*=

_{e}*L*+ 2

*l*and

_{e}*z*

_{0}=

*l*+

_{e}*ℓ*, and the transmitted flux should be calculated at the physical boundary, i.e.

_{t}*z*=

*l*+

_{e}*L*.

*R*and

*L*, are both much larger than the transport mean free path

*ℓ*(typically by a factor of about 10) [35

_{t}35. R. Elaloufi, R. Carminati, and J.-J. Greffet, “Diffusive-to-ballistic transition in dynamic light transmission through thin scattering slabs: a radiative transfer approach,” J. Opt. Soc. Am. A **21**, 1430–1437 (2004). [CrossRef]

*ℓ*= 5

_{t}*μ*m and

*L*=

*R*= 100

*μ*m. The statistics was made on a total of 10

^{7}random walkers (10 realizations of 10

^{6}random walkers).

16. D. Contini, F. Martelli, and G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation. I. Theory,” Appl. Opt. **36**, 4587–4599 (1997). [CrossRef] [PubMed]

^{−1}, is underestimated by about 38 % (Γ = 0.7045 ps

^{−1}). An excellent agreement is also observed on the total transmission through the cylinder, where Monte Carlo simulations yield

*T*= 0.049777 ± 0.000431 (95 % confidence interval) and Eq. (14)

*T*= 0.049483, while the theoretical prediction for a slab gives

*T*= 0.078125, that is overestimated by about 58 %. On this last point, we should point out that a very large number of terms in the sum over the sines and cosines is in general necessary to achieve convergence (here,

*max*[

*n*] = 20000), while much fewer are needed for the Bessel functions (here,

*max*[

*m*] = 10). In sum, this comparison with numerical calculations validates our theoretical model and proves very clearly the inadequacy of the diffusion model for slabs in such systems. Clearly, the deviation of the results between slab and cylinder configurations should diminish with increasing ratio

*R/L*.

## 4. Experiments and discussion

*L*was varied between about 1 and 10 mm and their radius

*R*was fixed to 3.5 mm. According to the mercury intrusion porosimetry measurements (Table 1), the samples are composed by 91 % of air (in the form of inclusions) and, thus, 9 % of mesoporous SiO

_{2}with permittivity around 2.1. From these considerations and using the Maxwell Garnett mixing rule [36

36. A. Sihvola, *Electromagnetic Mixing Formulae and Applications* (The Institution of Engineering and Technology, 1999). [CrossRef]

*n*&ap 1.04. This is expected to modify slightly the average energy velocity in the medium

_{eff}*v*=

*c*/

*n*, which enters the diffusion constant expression

_{eff}*D*, and yield a weak internal reflection coefficient

*r*&ap 0.059 in the expression of the extrapolated length

*l*[32

_{e}32. J. X. Zhu, D. J. Pine, and D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A **44**, 3948–3959 (1991). [CrossRef] [PubMed]

37. A. S. Gittings, R. Bandyopadhyay, and D. J. Durian, “Photon channelling in foams,” Europhys. Lett. **65**, 414–419 (2004). [CrossRef]

38. M. Schmiedeberg, M. F. Miri, and H. Stark, “Photon channelling in foams,” Eur. Phys. J. E **18**, 123–131 (2005). [CrossRef] [PubMed]

*D*=

*vℓ*/3, applies exclusively to disordered media exhibiting a statistically homogeneous distribution of scatterers. A more general formula for

_{t}*D*was derived recently for media containing arbitrary pore-size distributions [39

39. T. Svensson, K. Vynck, M. Grisi, R. Savo, M. Burresi, and D. S. Wiersma, “Holey random walks: Optics of heterogeneous turbid composites,” Phys. Rev. E **87**, 022120 (2013). [CrossRef]

*ℓ*to be smaller than the accuracy of our measurements. Nevertheless, we believe that this could be an interesting aspect to consider in future investigations.

_{t}*λ*= 1030 nm wavelength, 10 MHz repetition rate, 300 fs line width pulses delivered by a diode-pumped Ytterbium femtosecond oscillator from Amplitude systems (t-Pulse 200). The beam was focussed to a 1 mW, 100

*μ*m spot diameter on the slab; the transmission intensity was focussed on the entrance slit of the spectrograph.

18. E. Akkermans and G. Montambaux, *Mesoscopic Physics of Electrons and Photons* (Cambridge University Press, 2007). [CrossRef]

20. P. D. Garcia, R. Sapienza, J. Bertolotti, M. D. Martín, Á. Blanco, A. Altube, L. Vina, D. S. Wiersma, and C. López, “Resonant light transport through Mie modes in photonic glasses,” Phys. Rev. A **78**, 023823 (2008). [CrossRef]

*ℓ*= 53 ± 18

_{t}*μ*m and

*ℓ*= 1/

_{a}*μ*= 1.57 ± 0.21 mm (black dashed line), which is of the order of the samples size. Although the fit appears to be good, we remark that the deviation from the linear trend occurs when the thickness of the cylinder becomes comparable to its radius, a situation in which the lateral size of the sample simply cannot be neglected. This absorption length thus appears to account, at least partly, for the lateral “scattering losses”. This is verified by fitting the experimental data with the analytical expression for the total transmission through finite diffusive cylinders, Eq. (14), yielding

_{a}*ℓ*= 48 ± 0.4

_{t}*μ*m and

*ℓ*= 0.95 ± 0.35 m (red dashed line). This very long absorption length demonstrates that indeed the large decrease of the transmittance curve with sample thickness is a result of the finite lateral size of the SiO

_{a}_{2}(HIPE) monoliths and not to the material absorption. The curve expected for

*ℓ*= 48

_{t}*μ*m and no absorption is also shown for comparison (red solid line), confirming that the actual role of material absorption is minor.

*ℓ*and

_{t}*ℓ*with a single measurement [19

_{a}19. A. Z. Genack and J. M. Drake, “Relationship between Optical Intensity, Fluctuations and Pulse Propagation in Random Media,” Europhys. Lett. **11**, 4331–4336 (1990). [CrossRef]

20. P. D. Garcia, R. Sapienza, J. Bertolotti, M. D. Martín, Á. Blanco, A. Altube, L. Vina, D. S. Wiersma, and C. López, “Resonant light transport through Mie modes in photonic glasses,” Phys. Rev. A **78**, 023823 (2008). [CrossRef]

_{2}(HIPE) samples with thicknesses 3.2 mm (red stars) and 6.9 mm (blue circles). These profiles correspond to the convoluted responses of the instrument and of the monolith. Clearly, the samples exhibit a strongly multi-diffusive character, with photons being significantly delayed in the sample in both cases. As the thickness of the sample increases, so does the mean exit time of light. A small crack in the 6.9-mm thick sample (blue circles) caused a part of the excitation pulse to exit early, as observed by the shoulder at about 0.4 ns. These experimental temporal profiles have been compared to the predictions of the diffusion model in finite cylinders, Eq. (13), with the parameters determined previously, i.e.

*ℓ*= 48

_{t}*μ*m and

*ℓ*= 0.95 m. Fig. 3(c) shows the predicted curves (solid lines), which are the convolutions of Eq. (13) with the excitation pulse. Experimental data and theoretical predictions match remarkably well, further showing the possibility to retrieve transport parameters in cylindrically-shaped diffusive materials.

_{a}*L*considered in this study at a wavelength of 515 nm is shown in Fig. 3(d) (red stars). Fits of Eq. (15) to the data yield

*ℓ*= 48 ± 3

_{t}*μ*m and

*ℓ*= 1.2 ± 3.7 m, which are in very good agreement with the values obtained from the steady-state measurements (Fig. 3(b)). Again, a comparison with the curve expected for

_{a}*ℓ*= 48

_{t}*μ*m and no absorption highlights the negligible role of material absorption, explaining also the very large uncertainty on

*ℓ*. This therefore confirms the validity of our transport parameter recovery and of our interpretation of the results.

_{a}## 5. Conclusion

_{2}(HIPE) monoliths exhibiting a broad pore-size distribution centered around 19

*μ*m. These systems resemble certain biological tissues and thus, could be used as testbeds for optical spectroscopy techniques. We have developed a model of diffusion in finite cylinders to obtain analytical expressions for the steady-state and time-resolved transmission along the cylinder axis. These expressions have been validated by a comparison with Monte Carlo random walk simulations. Steady-state and time-resolved measurements have been performed on samples with varying thickness and the transport mean free path and material absorption length have been retrieved from fits of the experimental curves with the derived analytical expressions. Our results unambiguously show that the scattering losses on the lateral sides of the cylinder behave in a similar way as material absorption losses. Thus, neglecting the finite lateral sample size in the retrieval of the photon transport parameters can lead to a significant error in the estimation of the material absorption length. On the other hand, the problem is completely resolved when the analytical expressions for diffusion in finite cylinders are used. The recognition of this geometrical effect may be important in several practical cases, including biological tissue samples, pharmaceutical tablets, and macroporous solar cell photoanodes.

## References and links

1. | A. B. Davis and A. Marshak, “Solar radiation transport in the cloudy atmosphere: a 3D perspective on observations and climate impacts,” Rep. Prog. Phys. |

2. | S. Chandrasekhar, |

3. | G. Reich, “Near-infrared spectroscopy and imaging: Basic principles and pharmaceutical applications,” Adv. Drug Delivery Rev. |

4. | H. W. Siesler, Y. Ozaki, S. Kawata, and H. M. Heise, “ |

5. | V. V. Tuchin, |

6. | L. V. Wang and H.-i. Wu, |

7. | M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. |

8. | D. A. Benaron and D. K. Stevenson, “Optical time-of-flight and absorbance imaging of biologic media,” Science |

9. | T. Svensson, E. Alerstam, D. Khoptyar, J. Johansson, S. Folestad, and S. Andersson-Engels, “Near-infrared photon time-of-flight spectroscopy of turbid materials up to 1400 nm,” Rev. Sci. Instrum. |

10. | V. Backman, M. B. Wallace, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Müller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, S. Shapshay, T. Valdez, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, J. Van Dam, and M. S. Feld, “Detection of preinvasive cancer cells,” Nature |

11. | E. Alerstam and T. Svensson, “Observation of anisotropic diffusion of light in compacted granular porous materials,” Phys. Rev. E |

12. | T. Svensson, M. Andersson, L. Rippe, S. Svanberg, S. Andersson-Engels, J. Johansson, and S. Folestad, “VCSEL-based oxygen spectroscopy for structural analysis of pharmaceutical solids,” Appl. Phys. B |

13. | Z. Shi and C. A. Anderson, “Pharmaceutical applications of separation of absorption and scattering in near-infrared spectroscopy (NIRS),” J. Pharm. Sci. |

14. | C. M. Leroy, C. Olivier, T. Toupance, M. Abbas, L. Hirsch, S. Ravaine, and R. Backov, “One-pot easily-processed TiO |

15. | A. Ishimaru, |

16. | D. Contini, F. Martelli, and G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation. I. Theory,” Appl. Opt. |

17. | M. C. W. van Rossum and Th. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. |

18. | E. Akkermans and G. Montambaux, |

19. | A. Z. Genack and J. M. Drake, “Relationship between Optical Intensity, Fluctuations and Pulse Propagation in Random Media,” Europhys. Lett. |

20. | P. D. Garcia, R. Sapienza, J. Bertolotti, M. D. Martín, Á. Blanco, A. Altube, L. Vina, D. S. Wiersma, and C. López, “Resonant light transport through Mie modes in photonic glasses,” Phys. Rev. A |

21. | T. van der Beek, P. Barthelemy, P. M. Johnson, D. S. Wiersma, and A. Lagendijk, “Light transport through disordered layers of dense gallium arsenide submicron particles,” Phys. Rev. B |

22. | M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. |

23. | S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the Transmission Matrix in Optics: An Approach to the Study and Control of Light Propagation in Disordered Media,” Phys. Rev. Lett. |

24. | R. Savo, M. Burresi, T. Svensson, K. Vynck, and D. S. Wiersma, “Measuring the fractal dimension of an optical random walk,” arXiv:1312.5962. |

25. | N. Ghofraniha, I. Viola, A. Zacheo, V. Arima, G. Gigli, and C. Conti, “Transition from nonresonant to resonant random lasers by the geometrical confinement of disorder,” Opt. Lett. |

26. | F. Carn, A. Colin, M-.F. Achard, M. Pirot, H. Deleuze, and R. Backov, “Inorganic monoliths hierarchically textured via concentrated direct emulsion and micellar templates,” J. Mat. Chem. |

27. | D. Barby and Z. Haq, “Low density porous cross-linked polymeric materials and their preparation,” Eur. Patent Appl.60138 (1982). |

28. | M. J. Mooney, “The viscosity of a concentrated suspension of spherical particles,” J. Colloid. Interface Sci. |

29. | T. G. Mason, J. Bibette, and D. A. Weitz, “Yielding and Flow of Monodisperse Emulsions,” J. Colloid. Interface Sci. |

30. | M.-P. Aronson and M.-F. Petko, “Highly Concentrated Water-in-Oil Emulsions: Influence of Electrolyte on Their Properties and Stability,” J. Colloid Interface Sci. |

31. | C. J. Brinker and G. W. Scherer, in |

32. | J. X. Zhu, D. J. Pine, and D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A |

33. | L.-H. Wang, S. L. Jacques, and L.-Q. Zheng, ”Monte Carlo modeling of photon transport in multi-layered tissues,” Computer Methods and Programs in Biomedicine |

34. | M. Xu and R. R. Alfano, “Random walk of polarized light in turbid media,” Phys. Rev. Lett. |

35. | R. Elaloufi, R. Carminati, and J.-J. Greffet, “Diffusive-to-ballistic transition in dynamic light transmission through thin scattering slabs: a radiative transfer approach,” J. Opt. Soc. Am. A |

36. | A. Sihvola, |

37. | A. S. Gittings, R. Bandyopadhyay, and D. J. Durian, “Photon channelling in foams,” Europhys. Lett. |

38. | M. Schmiedeberg, M. F. Miri, and H. Stark, “Photon channelling in foams,” Eur. Phys. J. E |

39. | T. Svensson, K. Vynck, M. Grisi, R. Savo, M. Burresi, and D. S. Wiersma, “Holey random walks: Optics of heterogeneous turbid composites,” Phys. Rev. E |

**OCIS Codes**

(290.4210) Scattering : Multiple scattering

(290.5820) Scattering : Scattering measurements

(290.5825) Scattering : Scattering theory

**ToC Category:**

Materials

**History**

Original Manuscript: February 18, 2014

Manuscript Accepted: March 7, 2014

Published: March 24, 2014

**Virtual Issues**

Vol. 9, Iss. 6 *Virtual Journal for Biomedical Optics*

**Citation**

P. Gaikwad, S. Ungureanu, R. Backov, K. Vynck, and R. A. L. Vallée, "Photon transport in cylindrically-shaped disordered meso-macroporous materials," Opt. Express **22**, 7503-7513 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-7-7503

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### References

- A. B. Davis, A. Marshak, “Solar radiation transport in the cloudy atmosphere: a 3D perspective on observations and climate impacts,” Rep. Prog. Phys. 73, 026801 (2010). [CrossRef]
- S. Chandrasekhar, Radiative Transfer (Dover Publications, 2011).
- G. Reich, “Near-infrared spectroscopy and imaging: Basic principles and pharmaceutical applications,” Adv. Drug Delivery Rev. 57, 1109 (2005). [CrossRef]
- H. W. Siesler, Y. Ozaki, S. Kawata, H. M. Heise, “Near-Infrared Spectroscopy: Principles, Instruments, Applications” (Wiley, New York, 2008).
- V. V. Tuchin, Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis, 2 (SPIE, Bellingham, WA, 2007). [CrossRef]
- L. V. Wang, H.-i. Wu, Biomedical Optics: Principles and Imaging (Wiley, New York, 2007).
- 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]
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